Groupprops - User contributions [en]

General linear group over a finite field

2024-12-05T13:36:27Z

R-a-jones:

{{natural number-parametrized linear algebraic group}}

For the more general family of groups over *any* field, see [[General linear group over a field]].

==Definition==

===In terms of dimension (finite-dimensional case)===
Let <math>n</math> be a natural number and <math>k</math> a finite field. The '''general linear group''' of degree <math>n</math> over <math>k</math>, denoted <math>GL(n,k)</math>, is defined in the following equivalent ways:

* <math>GL(n,k)</math> is the group of all invertible <math>k</math>-linear maps from the vector space <math>k^n</math> to itself, under composition. In other words, it is the group of automorphisms of <math>k^n</math> as a <math>k</math>-vector space.
* <math>GL(n,k)</math> is the group of all invertible <math>n \times n</math> matrices with entries over <math>k</math>

==Arithmetic functions==

<math>q = p^a</math> here denotes the size of the finite field <math>k</math>.

{| class="sortable" border="1"
! Function !! Value !! Explanation
|-
| [[Order of a group|order]] || <math>(q^n-1)(q^n-q)...(q^n-q^{n-1})</math> || In order for the matrix to have non-zero determinant, the vector in the first column cannot be the zero vector, so we have <math>(q^n-1)</math> choices for such a vector. The vector in the second column must be linearly independent with the first column, giving <math>(q^n-q)</math> choices. In general, the vector in the ith column must be independent of the first, second, ..., i-1th column, giving <math>(q^n-q^{i-1})</math> choices. Make such a choice of vector for each column.
|}

==Particular cases==

{| class="wikitable" border="1"
!Size of field !! Order of matrices !! Common name for the general linear group !! Order of group !! Comment
|-
| 2 || 1 || [[Trivial group]] || <math>1</math> || Trivial
|-
| 3 || 1 || [[Cyclic group:Z2]] || <math>2</math> || [[group of prime order]]
|-
| 4 || 1 || [[Cyclic group:Z3]] || <math>3</math> || [[group of prime order]]
|-
| 5 || 1 || [[Cyclic group:Z4]] || <math>4 = 2^2</math> || [[cyclic group]]
|-
| 2 || 2 || [[Symmetric group:S3]] || <math>6 = 2 \cdot 3</math> || [[Supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]]
|-
| 3 || 2 || [[General linear group:GL(2,3)]] || <math>48 = 2^4 \cdot 3</math> || [[solvable group|solvable]] but not supersolvable
|-
| 4 || 2 || [[Alternating group:A5]] || <math>60 = 2^2 \cdot 3 \cdot 5</math> || [[simple non-abelian group]]
|-
| 5 || 2 || [[General linear group:GL(2,5)]] || <math>480 = 2^5 \cdot 3 \cdot 5</math> || not solvable, has a simple non-abelian [[subquotient]].
|-
| 2 || 3 || [[General linear group:GL(3,2)]] || <math>168 = 2^3 \cdot 3 \cdot 7</math> || [[simple non-abelian group]]
|-
|}

General linear group over a finite field

2024-12-05T13:35:39Z

R-a-jones: Created page with "{{natural number-parametrized linear algebraic group}} For the more general family of groups over *any* field, see General linear group over a field. ==Definition== ===In terms of dimension (finite-dimensional case)=== Let <math>n</math> be a natural number and <math>k</math> a finite field. The '''general linear group''' of degree <math>n</math> over <math>k</math>, denoted <math>GL(n,k)</math>, is defined in the following equivalent ways: * <math>GL(n,k)</math>..."

Engel group

2024-12-03T15:07:14Z

R-a-jones: /* Examples and counterexamples */

{{group property}}

==Definition==

A [[group]] <math>G</math> is termed an '''Engel group''' or '''nil group''' or '''nilgroup''', if, given any two elements <math>x,y \in G</math>, there exists a <math>n</math> such that the iterated [[commutator]]:

<math>[[ \dots [x,y],y],y],\dots],y] = e</math>

where <math>e</math> denotes the identity element, <math>[x,y] = xyx^{-1}y^{-1}</math> denotes the commutator of <math>x</math> and <math>y</math>, and <math>y</math> occurs <math>n</math> times.

If there exists a <math>n</math> that works for all pairs of elements of <math>G</math>, then we say that <math>G</math> is a <math>n</math>-Engel group. A <math>n</math>-Engel group, for some <math>n</math>, is termed a [[bounded Engel group]]. Note that sometimes the term ''Engel group'' is used for bounded Engel group.

Note if we instead define the commutator as <math>[x,y] = x^{-1}y^{-1}xy</math> we get an equivalent definition.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::locally nilpotent group]] || every [[finitely generated group|finitely generated]] subgroup is [[nilpotent group|nilpotent]] || || || {{intermediate notions short|Engel group|locally nilpotent group}}
|-
| [[Weaker than::nilpotent group]] || || || || {{intermediate notions short|Engel group|nilpotent group}}
|-
| [[Weaker than::bounded Engel group]] || || || || {{intermediate notions short|Engel group|bounded Engel group}}
|-
| [[Weaker than::2-locally nilpotent group]] || subgroup generated by two elements is always nilpotent || || || {{intermediate notions short|Engel group|2-locally nilpotent group}}
|}

==Examples and counterexamples==

===Finite groups===

* All the [[nilpotent group]]s, which are equivalent to [[locally nilpotent group]]s for finite groups, are Engel groups.

* The smallest non-Engel finite group is [[symmetric group:S3]]. To see that, consider <math>x = (1 \, 3)</math>, <math>y = (2 \, 3)</math>. Define <math>x_0 = x</math>, <math>x_i = [x_{i-1}, y]</math> for <math>i \geq 1</math> and you will see that none of <math>x_0, x_1, x_2, \dots</math> are the identity permutation. (The values of <math>x_0, x_1, x_2, \dots</math> depend on which definition of commutator you use, and which convention you take on the order of composition of permutations)

Engel group

2024-12-03T15:04:56Z

R-a-jones: /* Examples and counterexamples */

{{group property}}

==Definition==

A [[group]] <math>G</math> is termed an '''Engel group''' or '''nil group''' or '''nilgroup''', if, given any two elements <math>x,y \in G</math>, there exists a <math>n</math> such that the iterated [[commutator]]:

<math>[[ \dots [x,y],y],y],\dots],y] = e</math>

where <math>e</math> denotes the identity element, <math>[x,y] = xyx^{-1}y^{-1}</math> denotes the commutator of <math>x</math> and <math>y</math>, and <math>y</math> occurs <math>n</math> times.

If there exists a <math>n</math> that works for all pairs of elements of <math>G</math>, then we say that <math>G</math> is a <math>n</math>-Engel group. A <math>n</math>-Engel group, for some <math>n</math>, is termed a [[bounded Engel group]]. Note that sometimes the term ''Engel group'' is used for bounded Engel group.

Note if we instead define the commutator as <math>[x,y] = x^{-1}y^{-1}xy</math> we get an equivalent definition.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::locally nilpotent group]] || every [[finitely generated group|finitely generated]] subgroup is [[nilpotent group|nilpotent]] || || || {{intermediate notions short|Engel group|locally nilpotent group}}
|-
| [[Weaker than::nilpotent group]] || || || || {{intermediate notions short|Engel group|nilpotent group}}
|-
| [[Weaker than::bounded Engel group]] || || || || {{intermediate notions short|Engel group|bounded Engel group}}
|-
| [[Weaker than::2-locally nilpotent group]] || subgroup generated by two elements is always nilpotent || || || {{intermediate notions short|Engel group|2-locally nilpotent group}}
|}

==Examples and counterexamples==

===Finite groups===

* For finite groups, the Engel groups are precisely the [[nilpotent group]]s, which are equivalent to [[locally nilpotent group]]s for finite groups, are Engel groups.

* The smallest non-Engel finite group is [[symmetric group:S3]]. To see that, consider <math>x = (1 \, 3)</math>, <math>y = (2 \, 3)</math>. Define <math>x_0 = x</math>, <math>x_i = [x_{i-1}, y]</math> for <math>i \geq 1</math> and you will see that none of <math>x_0, x_1, x_2, \dots</math> are the identity permutation. (The values of <math>x_0, x_1, x_2, \dots</math> depend on which definition of commutator you use, and which convention you take on the order of composition of permutations)

Engel group

2024-12-03T14:28:50Z

R-a-jones:

{{group property}}

==Definition==

A [[group]] <math>G</math> is termed an '''Engel group''' or '''nil group''' or '''nilgroup''', if, given any two elements <math>x,y \in G</math>, there exists a <math>n</math> such that the iterated [[commutator]]:

<math>[[ \dots [x,y],y],y],\dots],y] = e</math>

where <math>e</math> denotes the identity element, <math>[x,y] = xyx^{-1}y^{-1}</math> denotes the commutator of <math>x</math> and <math>y</math>, and <math>y</math> occurs <math>n</math> times.

If there exists a <math>n</math> that works for all pairs of elements of <math>G</math>, then we say that <math>G</math> is a <math>n</math>-Engel group. A <math>n</math>-Engel group, for some <math>n</math>, is termed a [[bounded Engel group]]. Note that sometimes the term ''Engel group'' is used for bounded Engel group.

Note if we instead define the commutator as <math>[x,y] = x^{-1}y^{-1}xy</math> we get an equivalent definition.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::locally nilpotent group]] || every [[finitely generated group|finitely generated]] subgroup is [[nilpotent group|nilpotent]] || || || {{intermediate notions short|Engel group|locally nilpotent group}}
|-
| [[Weaker than::nilpotent group]] || || || || {{intermediate notions short|Engel group|nilpotent group}}
|-
| [[Weaker than::bounded Engel group]] || || || || {{intermediate notions short|Engel group|bounded Engel group}}
|-
| [[Weaker than::2-locally nilpotent group]] || subgroup generated by two elements is always nilpotent || || || {{intermediate notions short|Engel group|2-locally nilpotent group}}
|}

==Examples and counterexamples==

===Finite groups===

* All [[nilpotent group]]s, which are equivalent to [[locally nilpotent group]]s for finite groups, are Engel groups.

* The smallest non-Engel finite group is [[symmetric group:S3]]. To see that, consider <math>x = (1 \, 3)</math>, <math>y = (2 \, 3)</math>. Define <math>x_0 = x</math>, <math>x_i = [x_{i-1}, y]</math> for <math>i \geq 1</math> and you will see that none of <math>x_0, x_1, x_2, \dots</math> are the identity permutation. (The values of <math>x_0, x_1, x_2, \dots</math> depend on which definition of commutator you use, and which convention you take on the order of composition of permutations)

Engel group

2024-12-03T09:49:17Z

R-a-jones: /* Definition */

2-abelian group

2024-12-03T09:39:40Z

R-a-jones: Redirected page to N-abelian group

#redirect [[n-abelian group]]

N-abelian group

2024-12-03T09:37:59Z

R-a-jones: /* Examples */

{{natural number-parametrized group property}}
==Definition==

Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.

As noted below, [[n-abelian iff (1-n)-abelian]], so it suffices to restrict attention to <math>n</math> a positive integer.
===Alternative definitions===

See [[Alperin's structure theorem for n-abelian groups]].

==Facts==
===General facts===

<section begin="general facts"/>
* [[n-abelian iff (1-n)-abelian]]
* The set of <math>n</math> for which <math>G</math> is <math>n</math>-abelian is termed the [[exponent semigroup]] of <math>G</math>. It is a submonoid of the multiplicative monoid of integers.
* [[abelian implies n-abelian for all n]]
* [[n-abelian implies every nth power and (n-1)th power commute]]
* [[n-abelian implies n(n-1)-central]]
* [[n-abelian iff abelian (if order is relatively prime to n(n-1))]]
* [[nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center]]
* [[(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies p-power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism]]
* [[Characterization of exponent semigroup of a finite p-group]]
* [[Alperin's structure theorem for n-abelian groups]]
<section end="general facts"/>

===Particular values===

<section begin="particular values"/>
{| class="sortable" border="1"
! Value of <math>n</math> (note that the condition for <math>n</math> is the same as the condition for <math>1-n</math>)!! Characterization of <math>n</math>-abelian groups !! Proof !! Other related facts
|-
| 0 || all groups || obvious ||
|-
| 1 || all groups || obvious ||
|-
| 2 || [[abelian group]]s only || [[2-abelian iff abelian]] || [[endomorphism sends more than three-fourths of elements to squares implies abelian]]
|-
| -1 || [[abelian group]]s only || [[-1-abelian iff abelian]] ||
|-
| 3 || [[3-abelian group]] means: [[2-Engel group]] and [[derived subgroup]] has exponent dividing three || [[Levi's characterization of 3-abelian groups]] || [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is endomorphism implies class three]]
|-
| -2 || same as for 3-abelian || (based on [[n-abelian iff (1-n)-abelian]]) ||
|}
<section end="particular values"/>

==Relation with other properties==

===Weaker properties===

* [[n-nilpotent group]]
* [[n-solvable group]]

==Examples==

===Finite groups===

It follows immediately from [[Lagrange's theorem]] that a [[finite group]] <math>G</math> is <math>|G|</math>-abelian. If <math>|G|</math> is relatively prime to <math>n(n-1)</math>, then <math>G</math> is <math>n</math>-abelian if and only if it is [[abelian group|abelian]] ([[N-abelian iff abelian (if order is relatively prime to n(n-1))|article]])

We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially <math>n</math>-abelian for any given <math>n</math>.

{| class="sortable" border="1"
! <math>n</math>!! Non-abelian <math>n</math>-abelian groups.
|-
| 1 || all non-abelian finite groups
|-
| 2 || no non-abelian finite groups ([[2-abelian iff abelian]])
|-
| 3 || There are 10 3-abelian non-abelian finite groups with order at most 100 up to isomorphism: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
|-
| 4 || There are 231 4-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
|-
| 5 || There are 221 5-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
|-
| 6 || There are 86 6-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest example is [[symmetric group:S3]].
|}

N-abelian group

2024-12-03T09:33:48Z

R-a-jones: /* Finite groups */

{{natural number-parametrized group property}}
==Definition==

Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.

As noted below, [[n-abelian iff (1-n)-abelian]], so it suffices to restrict attention to <math>n</math> a positive integer.
===Alternative definitions===

See [[Alperin's structure theorem for n-abelian groups]].

==Facts==
===General facts===

<section begin="general facts"/>
* [[n-abelian iff (1-n)-abelian]]
* The set of <math>n</math> for which <math>G</math> is <math>n</math>-abelian is termed the [[exponent semigroup]] of <math>G</math>. It is a submonoid of the multiplicative monoid of integers.
* [[abelian implies n-abelian for all n]]
* [[n-abelian implies every nth power and (n-1)th power commute]]
* [[n-abelian implies n(n-1)-central]]
* [[n-abelian iff abelian (if order is relatively prime to n(n-1))]]
* [[nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center]]
* [[(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies p-power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism]]
* [[Characterization of exponent semigroup of a finite p-group]]
* [[Alperin's structure theorem for n-abelian groups]]
<section end="general facts"/>

===Particular values===

<section begin="particular values"/>
{| class="sortable" border="1"
! Value of <math>n</math> (note that the condition for <math>n</math> is the same as the condition for <math>1-n</math>)!! Characterization of <math>n</math>-abelian groups !! Proof !! Other related facts
|-
| 0 || all groups || obvious ||
|-
| 1 || all groups || obvious ||
|-
| 2 || [[abelian group]]s only || [[2-abelian iff abelian]] || [[endomorphism sends more than three-fourths of elements to squares implies abelian]]
|-
| -1 || [[abelian group]]s only || [[-1-abelian iff abelian]] ||
|-
| 3 || [[3-abelian group]] means: [[2-Engel group]] and [[derived subgroup]] has exponent dividing three || [[Levi's characterization of 3-abelian groups]] || [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is endomorphism implies class three]]
|-
| -2 || same as for 3-abelian || (based on [[n-abelian iff (1-n)-abelian]]) ||
|}
<section end="particular values"/>

==Relation with other properties==

===Weaker properties===

* [[n-nilpotent group]]
* [[n-solvable group]]

==Examples==

===Finite groups===

We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially <math>n</math>-abelian for any given <math>n</math>.

{| class="sortable" border="1"
! <math>n</math>!! Non-abelian <math>n</math>-abelian groups.
|-
| 1 || all non-abelian finite groups
|-
| 2 || no non-abelian finite groups ([[2-abelian iff abelian]])
|-
| 3 || There are 10 3-abelian non-abelian finite groups with order at most 100 up to isomorphism: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
|-
| 4 || There are 231 4-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
|-
| 5 || There are 221 5-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
|-
| 6 || There are 86 6-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest example is [[symmetric group:S3]].
|}

N-abelian group

2024-12-03T09:30:09Z

R-a-jones: /* Finite groups */

{{natural number-parametrized group property}}
==Definition==

Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.

As noted below, [[n-abelian iff (1-n)-abelian]], so it suffices to restrict attention to <math>n</math> a positive integer.
===Alternative definitions===

See [[Alperin's structure theorem for n-abelian groups]].

==Facts==
===General facts===

<section begin="general facts"/>
* [[n-abelian iff (1-n)-abelian]]
* The set of <math>n</math> for which <math>G</math> is <math>n</math>-abelian is termed the [[exponent semigroup]] of <math>G</math>. It is a submonoid of the multiplicative monoid of integers.
* [[abelian implies n-abelian for all n]]
* [[n-abelian implies every nth power and (n-1)th power commute]]
* [[n-abelian implies n(n-1)-central]]
* [[n-abelian iff abelian (if order is relatively prime to n(n-1))]]
* [[nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center]]
* [[(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies p-power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism]]
* [[Characterization of exponent semigroup of a finite p-group]]
* [[Alperin's structure theorem for n-abelian groups]]
<section end="general facts"/>

===Particular values===

<section begin="particular values"/>
{| class="sortable" border="1"
! Value of <math>n</math> (note that the condition for <math>n</math> is the same as the condition for <math>1-n</math>)!! Characterization of <math>n</math>-abelian groups !! Proof !! Other related facts
|-
| 0 || all groups || obvious ||
|-
| 1 || all groups || obvious ||
|-
| 2 || [[abelian group]]s only || [[2-abelian iff abelian]] || [[endomorphism sends more than three-fourths of elements to squares implies abelian]]
|-
| -1 || [[abelian group]]s only || [[-1-abelian iff abelian]] ||
|-
| 3 || [[3-abelian group]] means: [[2-Engel group]] and [[derived subgroup]] has exponent dividing three || [[Levi's characterization of 3-abelian groups]] || [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is endomorphism implies class three]]
|-
| -2 || same as for 3-abelian || (based on [[n-abelian iff (1-n)-abelian]]) ||
|}
<section end="particular values"/>

==Relation with other properties==

===Weaker properties===

* [[n-nilpotent group]]
* [[n-solvable group]]

==Examples==

===Finite groups===

We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially <math>n</math>-abelian for any given <math>n</math>.

{| class="sortable" border="1"
! <math>n</math>!! Non-abelian <math>n</math>-abelian groups.
|-
| 1 || all non-abelian finite groups
|-
| 2 || no non-abelian finite groups ([[2-abelian iff abelian]])
|-
| 3 || There are 10 3-abelian non-abelian finite groups with order at most 100 up to isomorphism: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
|-
| 4 || There are 231 4-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
|}

N-abelian group

2024-12-03T09:25:49Z

R-a-jones: /* Examples */

{{natural number-parametrized group property}}
==Definition==

Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.

As noted below, [[n-abelian iff (1-n)-abelian]], so it suffices to restrict attention to <math>n</math> a positive integer.
===Alternative definitions===

See [[Alperin's structure theorem for n-abelian groups]].

==Facts==
===General facts===

<section begin="general facts"/>
* [[n-abelian iff (1-n)-abelian]]
* The set of <math>n</math> for which <math>G</math> is <math>n</math>-abelian is termed the [[exponent semigroup]] of <math>G</math>. It is a submonoid of the multiplicative monoid of integers.
* [[abelian implies n-abelian for all n]]
* [[n-abelian implies every nth power and (n-1)th power commute]]
* [[n-abelian implies n(n-1)-central]]
* [[n-abelian iff abelian (if order is relatively prime to n(n-1))]]
* [[nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center]]
* [[(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies p-power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism]]
* [[Characterization of exponent semigroup of a finite p-group]]
* [[Alperin's structure theorem for n-abelian groups]]
<section end="general facts"/>

===Particular values===

<section begin="particular values"/>
{| class="sortable" border="1"
! Value of <math>n</math> (note that the condition for <math>n</math> is the same as the condition for <math>1-n</math>)!! Characterization of <math>n</math>-abelian groups !! Proof !! Other related facts
|-
| 0 || all groups || obvious ||
|-
| 1 || all groups || obvious ||
|-
| 2 || [[abelian group]]s only || [[2-abelian iff abelian]] || [[endomorphism sends more than three-fourths of elements to squares implies abelian]]
|-
| -1 || [[abelian group]]s only || [[-1-abelian iff abelian]] ||
|-
| 3 || [[3-abelian group]] means: [[2-Engel group]] and [[derived subgroup]] has exponent dividing three || [[Levi's characterization of 3-abelian groups]] || [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is endomorphism implies class three]]
|-
| -2 || same as for 3-abelian || (based on [[n-abelian iff (1-n)-abelian]]) ||
|}
<section end="particular values"/>

==Relation with other properties==

===Weaker properties===

* [[n-nilpotent group]]
* [[n-solvable group]]

==Examples==

===Finite groups===

We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially <math>n</math>-abelian for any given <math>n</math>.

{| class="sortable" border="1"
! <math>n</math>!! Non-abelian <math>n</math>-abelian groups.
|-
| 1 || all non-abelian finite groups
|-
| 2 || no non-abelian finite groups ([[2-abelian iff abelian]])
|-
| 3 || There are 10 3-abelian non-abelian finite groups with order less than 100 up to isomorphism: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
|}

N-abelian group

2024-12-03T09:25:09Z

R-a-jones: /* Finite groups */ fixed table

{{natural number-parametrized group property}}
==Definition==

Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.

As noted below, [[n-abelian iff (1-n)-abelian]], so it suffices to restrict attention to <math>n</math> a positive integer.
===Alternative definitions===

See [[Alperin's structure theorem for n-abelian groups]].

==Facts==
===General facts===

<section begin="general facts"/>
* [[n-abelian iff (1-n)-abelian]]
* The set of <math>n</math> for which <math>G</math> is <math>n</math>-abelian is termed the [[exponent semigroup]] of <math>G</math>. It is a submonoid of the multiplicative monoid of integers.
* [[abelian implies n-abelian for all n]]
* [[n-abelian implies every nth power and (n-1)th power commute]]
* [[n-abelian implies n(n-1)-central]]
* [[n-abelian iff abelian (if order is relatively prime to n(n-1))]]
* [[nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center]]
* [[(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies p-power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism]]
* [[Characterization of exponent semigroup of a finite p-group]]
* [[Alperin's structure theorem for n-abelian groups]]
<section end="general facts"/>

===Particular values===

<section begin="particular values"/>
{| class="sortable" border="1"
! Value of <math>n</math> (note that the condition for <math>n</math> is the same as the condition for <math>1-n</math>)!! Characterization of <math>n</math>-abelian groups !! Proof !! Other related facts
|-
| 0 || all groups || obvious ||
|-
| 1 || all groups || obvious ||
|-
| 2 || [[abelian group]]s only || [[2-abelian iff abelian]] || [[endomorphism sends more than three-fourths of elements to squares implies abelian]]
|-
| -1 || [[abelian group]]s only || [[-1-abelian iff abelian]] ||
|-
| 3 || [[3-abelian group]] means: [[2-Engel group]] and [[derived subgroup]] has exponent dividing three || [[Levi's characterization of 3-abelian groups]] || [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is endomorphism implies class three]]
|-
| -2 || same as for 3-abelian || (based on [[n-abelian iff (1-n)-abelian]]) ||
|}
<section end="particular values"/>

==Relation with other properties==

===Weaker properties===

* [[n-nilpotent group]]
* [[n-solvable group]]

==Examples==

===Finite groups===

We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially <math>n</math>-abelian for any given <math>n</math>.

{| class="sortable" border="1"
! <math>n</math>!! Non-abelian <math>n</math>-abelian groups.
|-
| 1 || all non-abelian finite groups
|-
| 2 || no non-abelian finite groups ([[2-abelian iff abelian]])
|-
| 3 || There are 10 3-abelian non-abelian finite groups with order less than 100: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
|}

N-abelian group

2024-12-03T09:24:47Z

R-a-jones: some examples

{{natural number-parametrized group property}}
==Definition==

Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.

As noted below, [[n-abelian iff (1-n)-abelian]], so it suffices to restrict attention to <math>n</math> a positive integer.
===Alternative definitions===

See [[Alperin's structure theorem for n-abelian groups]].

==Facts==
===General facts===

<section begin="general facts"/>
* [[n-abelian iff (1-n)-abelian]]
* The set of <math>n</math> for which <math>G</math> is <math>n</math>-abelian is termed the [[exponent semigroup]] of <math>G</math>. It is a submonoid of the multiplicative monoid of integers.
* [[abelian implies n-abelian for all n]]
* [[n-abelian implies every nth power and (n-1)th power commute]]
* [[n-abelian implies n(n-1)-central]]
* [[n-abelian iff abelian (if order is relatively prime to n(n-1))]]
* [[nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center]]
* [[(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies p-power map is endomorphism]]
* [[Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism]]
* [[Characterization of exponent semigroup of a finite p-group]]
* [[Alperin's structure theorem for n-abelian groups]]
<section end="general facts"/>

===Particular values===

<section begin="particular values"/>
{| class="sortable" border="1"
! Value of <math>n</math> (note that the condition for <math>n</math> is the same as the condition for <math>1-n</math>)!! Characterization of <math>n</math>-abelian groups !! Proof !! Other related facts
|-
| 0 || all groups || obvious ||
|-
| 1 || all groups || obvious ||
|-
| 2 || [[abelian group]]s only || [[2-abelian iff abelian]] || [[endomorphism sends more than three-fourths of elements to squares implies abelian]]
|-
| -1 || [[abelian group]]s only || [[-1-abelian iff abelian]] ||
|-
| 3 || [[3-abelian group]] means: [[2-Engel group]] and [[derived subgroup]] has exponent dividing three || [[Levi's characterization of 3-abelian groups]] || [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is endomorphism implies class three]]
|-
| -2 || same as for 3-abelian || (based on [[n-abelian iff (1-n)-abelian]]) ||
|}
<section end="particular values"/>

==Relation with other properties==

===Weaker properties===

* [[n-nilpotent group]]
* [[n-solvable group]]

==Examples==

===Finite groups===

We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially <math>n</math>-abelian for any given <math>n</math>.

{| class="sortable" border="1"
! <math>n</math>!! Non-abelian <math>n</math> abelian groups. !!
|-
| 1 || all non-abelian finite groups
|-
| 2 || no non-abelian finite groups ([[2-abelian iff abelian]])
|-
| 3 || There are 10 3-abelian non-abelian finite groups with order less than 100: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
|}

Symmetric group:S3

2024-12-03T00:39:16Z

R-a-jones: /* Other properties */

{{perspectives}}
[[Category:General affine groups]]
[[Category:Symmetric groups]]
[[Category:Dihedral groups]]
[[Importance rank::1| ]]
{{TOCright}}
{{particular group}}
==Definition==

===Verbal definitions===

The symmetric group <math>S_3</math> can be defined in the following equivalent ways:

* It is the [[member of family::symmetric group]][[member of family::symmetric group on finite set| ]] on a set of three elements, viz., the group of all permutations of a three-element set. In particular, it is a [[member of family::symmetric group of prime degree]] and [[member of family::symmetric group of prime power degree]].
* It is the {{dihedral group}} of order six (degree three), viz., the group of (not necessarily orientation-preserving) symmetries of the equilateral triangle.
* It is the [[member of family::special linear group]] of [[member of family::special linear group of degree two|degree two]] <math>SL(2,2)</math> over the [[member of family::field:F2|field of two elements]]. It turns out that, because of the nature of the prime two, it is also the [[member of family::projective special linear group]] of [[member of family::projective special linear group of degree two|degree two]] <math>PSL(2,2)</math>, the [[member of family::general linear group]] of [[member of family::general linear group of degree two|degree two]] <math>GL(2,2)</math>, and the [[member of family::projective general linear group]] of [[member of family::projective general linear group of degree two|degree two]] <math>PGL(2,2)</math>.
* It is the [[member of family::general affine group]] of [[general affine group of degree one|degree one]] over the [[field:F3|field of three elements]], i.e., <math>GA(1,3)</math> (sometimes also written as <math>AGL(1,3)</math>).
* It is the [[member of family::general semilinear group]] of [[member of family::general semilinear group of degree one|degree one]] over the [[field:F4|field of four elements]], i.e., <math>\Gamma L(1,4)</math>.
* It is the [[member of family::von Dyck group]] with parameters <math>(2,2,3)</math>, and in particular, is a [[member of family::Coxeter group]]. In particular, it has the presentation (where <math>e</math> denotes the identity element):

<math>\langle a,b,c \mid a^2 = b^2 = c^3 = abc = e \rangle</math>.

In the Coxeter language, this is written as:

<math>\langle s_1, s_2 \mid s_1^2 = s_2^2 = (s_1s_2)^3 = e \rangle</math>.

===Multiplication table===

We portray elements as permutations on the set <math>\{ 1,2,3 \}</math> using the [[cycle decomposition]]. The ''row element is multiplied on the left and the column element on the right'', with the assumption of ''functions written on the left''. ''This means that the column element is applied first and the row element is applied next''.

{{#lst:element structure of symmetric group:S3|multiplication table}}

If we used the opposite convention (i.e., functions written on the right), the row element is to be multiplied on the right and the column element on the left.

Here is the multiplication table where we use the [[one-line notation]] for permutations, where, as in the previous multiplication table, the column permutation is applied first and then the row permutation. Thus, with the left action convention, the row element
is multiplied on the left and the column element on the right:

{{#lst:element structure of symmetric group:S3|multiplication table in one-line notation}}

==Families==

The symmetric group on three elements is part of some important families:

{| class="sortable" border="1"
! Generic name for family member !! Definition !!Parametrization of family !! Parameter value(s) for this member !! Other members !! Comments
|-
|[[member of family::symmetric group on finite set]] <math>S_n</math> || group of all permutations on a finite set || by a nonnegative integer <math>n</math>, denoting size of set acted on || <math>n = 3</math>, so the group is <math>S_3</math> || {{#ask: [[member of family::symmetric group on finite set]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::Coxeter group]] || has a presentation of a particular form || Coxeter matrix describing the presentation || || {{#ask: [[member of family::Coxeter group]]|limit = 0|searchlabel = click here for a list}}|| [[symmetric groups on finite sets are Coxeter groups]]
|-
| [[member of family::dihedral group]] <math>D_{2n}</math> || semidirect product of a cyclic group and a two-element group acting via the inverse map || by a positive integer <math>n</math> that's half the order || <math>n = 3</math>, so the group is <math>D_6</math> || {{#ask: [[member of family::dihedral group]]|limit = 0|searchlabel = click here for a list}}||
|-
| [[member of family::general affine group]] <math>GA(n,F)</math>|| semidirect product of a vector space over a field with the general linear group acting on that vector space || name of field <math>F</math>, degree <math>n</math>(i.e., dimension of vector space). For a finite field, we may also write the group as <math>GA(n,q)</math> where <math>q</math> is the size of the field || [[field:F3]] (size <math>q = 3</math>), [[general affine group of degree one over a finite field|degree one]], so the group is <math>GA(1,\mathbb{F}_3)</math> or <math>GA(1,3)</math> || {{#ask: [[member of family::general affine group of degree one over a finite field]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::general linear group]] <math>GL(n,F)</math> || [[general linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field. || [[field:F2]] (size <math>q = 2</math>), [[member of family::general linear group of degree two|degree two]], so the group is <math>GL(2,\mathbb{F}_2)</math> or <math>GL(2,2)</math> || {{#ask: [[member of family::general linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::projective general linear group]] <math>PGL(n,F)</math> || [[projective general linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field. || [[field:F2]] (size <math>q = 2</math>), [[member of family::projective general linear group of degree two|degree two]], so the group is <math>PGL(2,\mathbb{F}_2)</math> or <math>PGL(2,2)</math> || {{#ask: [[member of family::projective general linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::special linear group]] <math>SL(n,F)</math> || [[special linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field. || [[field:F2]] (size <math>q = 2</math>), [[member of family::special linear group of degree two|degree two]], so the group is <math>SL(2,\mathbb{F}_2)</math> or <math>SL(2,2)</math> || {{#ask: [[member of family::special linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::projective special linear group]] <math>PSL(n,F)</math> || [[projective special linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field. || [[field:F2]] (size <math>q = 2</math>), [[member of family::projective special linear group of degree two|degree two]], so the group is <math>PSL(2,\mathbb{F}_2)</math> or <math>PSL(2,2)</math> || {{#ask: [[member of family::special linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::general semilinear group]] <math>\Gamma L(n,F)</math> || semidirect product of general linear group and automorphism group of base field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field. || [[field:F4]] (size <math>q = 4</math>), [[general semilinear group of degree one|degree one]], so the group is <math>\Gamma L(1,\mathbb{F}_4) = \Gamma L(1,4)</math> || {{#ask: [[member of family::general semilinear group]]|limit = 0|searchlabel = click here for a list}} ||
|}

==Elements==

{{further|[[Element structure of symmetric group:S3]]}}

===Conjugacy class structure===

As for any [[symmetric group]], [[cycle type determines conjugacy class]]. The cycle types, in turn, are parametrized by the unordered integer partitions of <math>3</math>. The conjugacy classes are described below.

{{#lst:element structure of symmetric group:S3|conjugacy class structure}}

This group is one of three finite groups with the property that any two elements of the same order are conjugate. The other two are the [[cyclic group of order two]] and the [[trivial group]].

For an interpretation of the conjugacy class structure based on the other equivalent definitions of the group, visit [[Element structure of symmetric group:S3#Conjugacy class structure]].
===Automorphism class structure===

The classification of elements upto automorphism is the same as that upto conjugation; this is because the symmetric group on three elements is a [[complete group]]: a [[centerless group]] where every automorphism is inner.

==Arithmetic functions==

===Basic arithmetic functions===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value order|6}} || as [[symmetric group]] <math>\! S_n, n = 3:</math>, <math>\! n! = 3! = 3 \cdot 2 \cdot 1 = 6</math><br>as [[general linear group of degree two]] <math>\! GL(2,q), q = 2:</math> <math>\! (q^2 - 1)(q^2 - q) = (2^2 - 1)(2^2 - 2) = 6</math><br>as [[general affine group of degree one]] <math>GA(1,q)</math>, <math>q = 3</math>: <math>q(q - 1) = 3 \cdot 2 = 6</math><br>as [[dihedral group]] <math>D_{2n}, n = 3</math>: <math>2 \cdot 3 = 6</math><br>as [[general semilinear group of degree one]] <math>\Gamma L(1,q), q = 4, q = p^r, p = 2, r = 2</math>: <math>r(q - 1) = 2(4 - 1) = 2(3) = 6</math><br><br>For more information, see [[element structure of symmetric group:S3#Order computation]]
|-
| {{arithmetic function value given order|exponent of a group|6|6}} || Elements of order <math>2</math> and <math>3</math>.<br>as [[symmetric group]] <math>\! S_n, n = 3</math>: <math>\! \operatorname{lcm} \{ 1,2, \dots, n \} = \operatorname{lcm} \{ 1,2,3 \} = 6</math><br>As [[general linear group of degree two]] <math>GL(2,q), q = 2</math>, underlying prime <math>p = 2</math>: <math>p(q^2 - 1) = 2(2^2 - 1) = 2(3) = 6</math><br>As [[general affine group of degree one]] <math>GA(1,q)</math>, <math>q = 3</math>, underlying prime <math>p = 3</math>: <math>p(q - 1) = 3(3 - 1) = 3(2) = 6</math><br>as [[dihedral group]] <math>D_{2n}, n = 3</math>: <math>\operatorname{lcm} \{ 2,n \} = \operatorname{lcm} \{ 2, 3 \} = 6</math>
|-
| {{arithmetic function value given order|derived length|2|6}} || Cyclic subgroup of order three is abelian, has abelian quotient.
|-
| {{arithmetic function value given order|minimum size of generating set|2|6}} || <math>(1,2), (1,2,3)</math><br>As [[symmetric group on a finite set]]: 2 (see [[symmetric group on a finite set is 2-generated]])
|-
| {{arithmetic function value given order|subgroup rank of a group|2|6}} || All proper subgroups are cyclic.
|-
| {{arithmetic function value given order|max-length of a group|2|6}} || Subgroup series going through subgroup of order two or three.
|}

===Arithmetic functions of an element-counting nature===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value given order|number of conjugacy classes|3|6}} || The three classes are the identity element, the transpositions, and the 3-cycles.<br>As <matH>S_n, n =3 </math>: [[number of unordered integer partitions]] of 3, equals 3<br>As <math>GL(2,q), q = 2</math>: <matH>q^2 - 1 = 2^2 - 1 = 3</math><br>As <math>D_{2n}, n = 3</math>: <math>(n + 3)/2 = (3 + 3)/2 = 3</math><br>As <math>GA(1,q), q = 3</math>: <math>q = 3</math><br>As <math>\Gamma L(1,p^2), p = 2</math>: <math>(p^2 + 3p - 4)/2 = (2^2 + 3 \cdot 2 - 4)/2 = 3</math><br><br>For a more elaborate explanation of these formulas, see [[element structure of symmetric group:S3#Number of conjugacy classes]]
|-
| {{arithmetic function value given order|number of equivalence classes under real conjugacy|3|6}} || Same as the number of conjugacy classes, because the group is an [[ambivalent group]].
|-
|{{arithmetic function value given order|number of conjugacy classes of real elements|3|6}} || Same as the number of conjugacy classes, because the group is an [[ambivalent group]].
|-
|{{arithmetic function value given order|number of equivalence classes under rational conjugacy|3|6}} || Same as the number of conjugacy classes, because the group is a [[rational group]].
|-
|{{arithmetic function value given order|number of conjugacy classes of rational elements|3|6}} || Same as the number of conjugacy classes, because the group is a [[rational group]].
|}

===Arithmetic functions of a subgroup-counting nature===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value|number of subgroups|6}} || || See [[subgroup structure of symmetric group:S3]]
|-
| {{arithmetic function value|number of conjugacy classes of subgroups|4}} || ||
|-
| {{arithmetic function value given order|number of normal subgroups|3|6}} ||
|-
| {{arithmetic function value|number of automorphism classes of subgroups|4}} || ||
|}

===Lists of numerical invariants===

{| class="sortable" border="1"
! List !! Value !! Explanation/comment
|-
| [[conjugacy class size set|conjugacy class sizes]] || 1,2,3 || See [[cycle type determines conjugacy class]], [[element structure of symmetric group:S3]], [[element structure of symmetric groups]]
|-
| [[order statistics]] || <math>1 \mapsto 1, 2 \mapsto 3, 3 \mapsto 2</math> ||
|-
| [[degrees of irreducible representations]] || 1,1,2 || See [[linear representation theory of symmetric group:S3]], [[linear representation theory of symmetric groups]]
|-
| orders of subgroups || 1,2,2,2,3,6 || See [[subgroup structure of symmetric group:S3]]
|-
|}

==Endomorphisms==

{{further|[[Endomorphism structure of symmetric group:S3]]}}

===Automorphisms===

Since <math>S_3</math> is a complete group, it is isomorphic to its automorphism group, where each element of <math>S_3</math> acts on <math>S_3</math> by conjugation. In fact, for <math>n \ne 2,6</math>, the symmetric group <math>S_n</math> is a complete group. {{further|[[Symmetric groups on finite sets are complete]]}}

==Group properties==

===Important properties===

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[Dissatisfies property::Abelian group]] || No || <math>(1,2)</math> and <math>(2,3)</math> don't commute || Smallest non-abelian group
|-
|[[Dissatisfies property::Nilpotent group]] || No || [[Centerless group|Centerless]]: The [[center]] is trivial || Smallest non-nilpotent group
|-
|[[Satisfies property::Metacyclic group]] || Yes || Cyclic normal subgroup of order three, cyclic quotient of order two ||
|-
|[[Satisfies property::Supersolvable group]] || Yes || [[Metacyclic implies supersolvable]] ||
|-
|[[Satisfies property::Solvable group]] || Yes || Metacyclic implies solvable ||
|}

===Other properties===

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[Satisfies property::T-group]] || Yes || ||
|-
|[[Satisfies property::Monolithic group]] || Yes|| Unique minimal normal subgroup of order three ||
|-
|[[Satisfies property::One-headed group]] || Yes|| Unique maximal normal subgroup of order three ||
|-
|[[Satisfies property::Jordan-unique group]] || Yes || There is a unique composition series ||
|-
|[[Satisfies property::SC-group]] || Yes || Every subgroup of it is a [[C-group]] || [[C-group]] means that every subgroup is permutably complemented
|-
|[[Satisfies property::Rational-representation group]] || Yes || [[Symmetric groups are rational-representation]] ||
|-
|[[Satisfies property::Rational group]] || Yes || [[Symmetric groups are rational]] || Also see [[classification of rational dihedral groups]]
|-
|[[Satisfies property::Ambivalent group]] || Yes || [[Symmetric groups are ambivalent]] ||
|-
|[[Satisfies property::Complete group]] || Yes || [[Symmetric groups are complete]], except degrees <math>2,6</math> ||
|-
|[[Satisfies property::Group isomorphic to its automorphism group]] || Yes || || Being a [[complete group]] is a stronger property
|-
|[[Satisfies property::Frobenius group]] || Yes || Frobenius kernel is alternating group, complement is any subgroup of order two. || Frobenius group on account of being <math>GA(1,3)</math>.
|-
|[[Satisfies property::Camina group]] || Yes || ||
|-
|[[Satisfies property::Z-group]] || Yes || Both the 2-Sylow subgroup ([[S2 in S3]]) and the 3-Sylow subgroup ([[A3 in S3]]) are cyclic. ||
|-
|[[Satisfies property::Schur-trivial group]] || Yes || [[Schur multiplier of Z-group is trivial]] ||
|-
|[[Dissatisfies property::2-Engel group]] || No || ||
|}

==Subgroups==

[[Image:S3latticeofsubgroups.png|500px]]

{{further|[[Subgroup structure of symmetric group:S3]]}}

{{#lst:subgroup structure of symmetric group:S3|summary}}

===Subgroup-defining functions and associated quotient-defining functions===

{{#lst:subgroup structure of symmetric group:S3|sdf summary}}

==Linear representation theory==

{{further|[[Linear representation theory of symmetric group:S3]]}}

===Summary===

{{#lst:linear representation theory of symmetric group:S3|summary}}

===Character table===

{{#lst:linear representation theory of symmetric group:S3|character table}}

==Distinguishing features==

===Smallest of its kind===

* This is the unique non-abelian group of smallest order. All groups of order up to <math>5</math>, and all other groups of order <math>6</math>, are abelian.
* This is the unique non-nilpotent group of smallest order. All groups of order up to <math>5</math>, and all other groups of order <math>6</math>, are nilpotent.
* This is the unique smallest nontrivial [[complete group]].

==GAP implementation==

{{access GAP implementation online using SAGE|id=5012}}

{{GAP ID|6|1}}

===Other descriptions===

{| class="sortable" border="1"
! Description !! Functions used
|-
| <tt>SymmetricGroup(3)</tt> || [[GAP:SymmetricGroup|SymmetricGroup]]
|-
| <tt>DihedralGroup(6)</tt> || [[GAP:DihedralGroup|DihedralGroup]]
|-
| <tt>GL(2,2)</tt> || [[GAP:GL|GL]]
|}

Dihedral group:D8

2024-12-03T00:36:51Z

R-a-jones: /* Other properties */

{{TOCright}}
{{particular group}}
[[Importance rank::2| ]]
[[Category:Dihedral groups]]
==Definition==

===Definition by presentation===

The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or ''the dihedral group of degree four'' (since its natural action is on four elements), or sometimes the ''octic group'', is defined by the following [[presentation of a group|presentation]], with <math>e</math> denoting the identity element:

<math>\langle x,a \mid a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math>

Here, the element <math>a</math> is termed the ''rotation'' or the ''generator of the cyclic piece'' and <math>x</math> is termed the ''reflection''.

{{quotation|Confused about presentations in general or this one in particular? If you're new to this stuff, check out [[constructing dihedral group:D8 from its presentation]]. Sophisticated group theorists need simply recall that [[presentation of semidirect product is disjoint union of presentations plus action by conjugation relations]]}}

===Geometric definition===

The '''dihedral group''' <math>D_8</math> (also called <math>D_4</math>) is defined as the group of all symmetries of the square (the regular 4-gon). This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by <math>a</math>) and has four ''reflections'' each being an [[involution]]: reflections about lines joining midpoints of opposite sides, and reflections about diagonals.

===Definition as a permutation group===

{{further|[[D8 in S4]]}}

The group is (up to isomorphism) the subgroup of the symmetric group on <math>\{ 1,2,3,4 \}</math> given by:

<math>\! \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,3), (2,4), (1,4)(2,3), (1,2)(3,4) \}</math>

This can be related to the geometric definition by thinking of <math>1,2,3,4</math> as the vertices of the square and considering an element of <math>D_8</math> in terms of its induced action on the vertices. It relates to the presentation via setting <math>a = (1,2,3,4)</math> and <math>x = (1,3)</math>.

===Multiplication table===

Here, <math>e</math> denotes the identity element, <math>a</math> is an element of order 4, and <math>x</math> is an element of order two that isn't equal to <math>a^2</math>, as in the above presentation.

{{#lst:element structure of dihedral group:D8|multiplication table}}

===Other definitions===

The dihedral group can be described in the following ways:

# The [[dihedral group]] of order eight.
# The [[generalized dihedral group]] corresponding to [[cyclic group:Z4|the cyclic group of order four]].
# The [[holomorph of a group|holomorph]] of the [[cyclic group:Z4|cyclic group of order four]].
# The [[external wreath product]] of the cyclic group of order two with the cyclic group of order two, acting via the regular action.
# The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S4|symmetric group on four letters]].
# The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S5|symmetric group on five letters]].
# The <math>2</math>-[[Sylow subgroup]] of the [[alternating group:A6|alternating group on six letters]].
# The [[member of family::unitriangular matrix group of degree three]] <math>UT(3,2)</math> over [[field:F2]], <math>2</math>-[[Sylow subgroup]] of [[projective special linear group:PSL(3,2)|PSL(3,2)]].
# The [[member of family::extraspecial group]] of order <math>2^3</math> and type '+'.

==Position in classifications==

{| class="sortable" border="1"
! Type of classification !! Name in that classification
|-
| GAP ID || (8,3), i.e., the third among the groups of order 8
|-
| Hall-Senior number || (8,4), i.e., 4 among groups of order 8
|-
| Hall-Senior symbol || <math>8\Gamma_2a_1</math>
|}

==Elements==

{{further|[[element structure of dihedral group:D8]]}}
{{#lst:element structure of dihedral group:D8|elements}}

Below is the conjugacy and automorphism class structure:

{{#lst:element structure of dihedral group:D8|conjugacy and automorphism class structure}}

==Arithmetic functions==

===Basic arithmetic functions===

{{compare and contrast arithmetic functions|order = 8}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|8}} || As a semidirect product of <math>\Z_4</math> and <math>\Z_2</math>: the order is the product of the orders of <math>\Z_4</math> and <math>\Z_2</math>, which is <math>4 \times 2 = 8</math> <br><br> As a wreath product of <math>\Z_2</math> and <math>\Z_2</math>: the order is <math>2^2 \cdot 2 = 8</math>
|-
| {{arithmetic function value order p-log etc|3}}
|-
| {{arithmetic function value exponent given order|4|8}} || As a dihedral group: the dihedral group of order <math>2n</math> has exponent equal to <math>\operatorname{lcm} \{ n,2 \}</math>.
|-
| [[prime-base logarithm of exponent]] || [[arithmetic function value::prime-base logarithm of exponent;2|2]] || ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|8|3}} || The [[derived subgroup]] is <math>\{ e, a^2 \}</math> -- and it is the same as the [[center]]. See [[center of dihedral group:D8]]. Also see [[element structure of dihedral group:D8#Commutator map]]
|-
| {{arithmetic function value given order and p-log|derived length|2|8|3}} || The [[derived subgroup]] is <math>\{ e, a^2 \}</math>, which is abelian. See [[center of dihedral group:D8]].
|-
| {{arithmetic function value given order and p-log|Frattini length|2|8|3}} || The [[Frattini subgroup]] is <math>\{ e, a^2 \}</math>, which is of prime order, hence its Frattini subgroup is trivial.
|-
| [[Fitting length]] || [[arithmetic function value::Fitting length;1|1]] || || All groups of prime power order are nilpotent, hence have Fitting length 1.
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|8|3}} || Generator of cyclic subgroup of order four and element of order two outside.
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|2|8|3}} || All proper subgroups are cyclic or [[Klein four-group]]s.
|-
| {{arithmetic function value given order and p-log|rank of a p-group|2|8|3}} || There exist Klein four-subgroups.
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|2|8|3}} || There exist normal Klein four-subgroups.
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|1|8|3}} || All abelian characteristic subgroups are cyclic.
|}

===Arithmetic functions of an element-counting nature===

{{further|[[element structure of dihedral group:D8]]}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value !! GAP verification (set <tt>G := DihedralGroup(8);</tt>) -- See more at [[#GAP implementation]]
|-
| {{arithmetic function value given order|number of conjugacy classes|5|8|3}} || As dihedral group <math>D_{2n}</math>, <math>n</math> even: <br><math>\! (n + 6)/2 = (4 + 6)/2 = 5</math>. See [[element structure of dihedral groups]] and [[element structure of dihedral group:D8]]<br>As unitriangular matrix group <math>UT(3,q), q = 2</math>:<br><math>q^2 + q - 1 = 2^2 + 2 - 1 = 5</math><br>See [[element structure of unitriangular matrix group of degree three over a finite field]] || {{GAP verify list length|ConjugacyClasses}}
|-
| {{arithmetic function value given order|number of equivalence classes under real conjugacy|5|8|3}} || Same as number of conjugacy classes, because the group is an [[ambivalent group]]. See [[dihedral groups are ambivalent]] ||
|-
| {{arithmetic function value given order|number of conjugacy classes of real elements|5|8|3}} || Same as number of conjugacy classes, because the group is an [[ambivalent group]]. See [[dihedral groups are ambivalent]] ||
|-
| {{arithmetic function value given order|number of equivalence classes under rational conjugacy|5|8|3}} || Same as number of conjugacy classes, because the group is a [[rational group]]. || {{GAP verify list length|RationalClasses}}
|-
| {{arithmetic function value given order|number of conjugacy classes of rational elements|5|8|3}} || Same as number of conjugacy classes, because the group is a [[rational group]]. ||
|}

===Arithmetic functions of a subgroup-counting nature===

{{further|[[subgroup structure of dihedral group:D8]]}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation !! GAP verification (set <tt>G := DihedralGroup(8);</tt>) -- See more at [[#GAP verification]]
|-
| {{arithmetic function value|number of subgroups|10}} || || As a dihedral group <math>\! D_{2n}, n = 4</math> number of subgroups is <math>\! d(n) + \sigma(n) = d(4) + \sigma(4) = 3 + 7 = 10</math>, where <math>d</math> is the divisor count function and <math>\sigma</math> is the divisor sum function. See [[subgroup structure of dihedral group:D8]], [[subgroup structure of dihedral groups]] || {{GAP verify list length|Subgroups}}
|-
| {{arithmetic function value|number of conjugacy classes of subgroups|8}} || || See [[subgroup structure of dihedral groups]], [[subgroup structure of dihedral group:D8]] || {{GAP verify list length|ConjugacyClassesSubgroups}}
|-
| {{arithmetic function value given order|number of normal subgroups|6|8}} || See [[subgroup structure of dihedral groups]], [[subgroup structure of dihedral group:D8#Lattice of normal subgroups]] || {{GAP verify list length|NormalSubgroups}}
|-
| {{arithmetic function value|number of automorphism classes of subgroups|6}} || || ||
|-
| {{arithmetic function value|number of characteristic subgroups|4}} || || || {{GAP verify list length|CharacteristicSubgroups}}
|}

===Lists of numerical invariants===

{| class="sortable" border="1"
! List !! Value !! Explanation/comment
|-
| [[conjugacy class size set|conjugacy class sizes]] || 1,1,2,2,2 || Two central elements, all others in conjugacy classes of size two. See [[element structure of dihedral group:D8]] and [[element structure of dihedral groups]].
|-
| sizes of orbits under automorphism group || 1,1,2,4 || Two central elements, one conjugacy class of elements of order four, one orbit of size four, comprising two conjugacy classes of size, with all elements non-central of order two.
|-
| [[order statistics]] || <math>1 \mapsto 1, 2 \mapsto 5, 4 \mapsto 2</math> || Of the five elements of order two, one is central. The other four are automorphic to each other. See [[element structure of dihedral group:D8]] and [[element structure of dihedral groups]]
|-
| [[degrees of irreducible representations]] || <math>1,1,1,1,2</math> || See [[linear representation theory of dihedral group:D8]]
|-
| orders of subgroups || <math>1,2,2,2,2,2,4,4,4,8</math> || See [[subgroup structure of dihedral group:D8]]
|}

===Action-based/automorphism group realization invariants===

{| class="sortable" border="1"
! Function !! Value !! Explanation
|-
| minimum degree of faithful representation || 2 ||
|-
| minimum degree of nontrivial irreducible representation || 2 ||
|-
| smallest size of set with faithful action || 4 ||
|-
| smallest size of set with faithful transitive action || 4 ||
|-
| [[symmetric genus]] || ? ||
|}

==Group properties==

{{compare and contrast group properties|order=8}}

===Important properties===

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
| {{group properties because p-group}}
|-
|[[Dissatisfies property::abelian group]] || No || <math>a</math> and <math>x</math> don't commute || Smallest non-abelian [[satisfies property::group of prime power order]]
|-
|[[Dissatisfies property::T-group]] || No || <math>\langle x \rangle \triangleleft \langle a^2,x \rangle</math>, which is normal, but <math>\langle x \rangle</math> is not normal || Smallest example for [[normality is not transitive]].
|-
|[[Satisfies property::monolithic group]] || Yes|| Unique minimal normal subgroup of order two ||
|}

===Other properties===

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[Dissatisfies property::one-headed group]] || No || Three distinct maximal normal subgroups of order four ||
|-
|[[Dissatisfies property::SC-group]] || No || ||
|-
|[[Satisfies property::ACIC-group]] || Yes || Every [[automorph-conjugate subgroup]] is [[characteristic subgroup|characteristic]] ||
|-
|[[satisfies property::algebra group]] || Yes || It is isomorphic to the [[unitriangular matrix group of degree three]] over [[field:F2]], which is clearly an algebraic group. ||
|-
| [[Satisfies property::ambivalent group]] || Yes || [[dihedral groups are ambivalent]] || Also see [[generalized dihedral groups are ambivalent]]
|-
|[[Satisfies property::rational group]] || Yes || Any two elements that generate the same cyclic group are conjugate || Thus, all characters are integer-valued.
|-
|[[Satisfies property::rational-representation group]] || Yes || All representations over characteristic zero are realized over the rationals. || Contrast with [[quaternion group]], that is rational but not rational-representation.
|-
| {{group properties because extraspecial}}
|-
| [[Satisfies property::maximal class group]] || Yes || ||
|-
|[[Dissatisfies property::Frobenius group]] || No || Frobenius groups are centerless, and this group isn't ||
|-
|[[Satisfies property::Camina group]] || Yes || [[extraspecial implies Camina]] ||
|-
|[[Satisfies property::group in which every element is automorphic to its inverse|Every element is automorphic to its inverse]] || Yes || Follows from being an [[ambivalent group]] ||
|-
|[[Satisfies property::group in which any two elements generating the same cyclic subgroup are automorphic|any two elements generating the same cyclic subgroup are automorphic]] || Yes || ||
|-
|[[Dissatisfies property::group in which every element is order-automorphic|every element is order-automorphic]] || No || ||
|-
|[[Satisfies property::directly indecomposable group]] || Yes || ||
|-
|[[Satisfies property::centrally indecomposable group]] || Yes || ||
|-
|[[Dissatisfies property::splitting-simple group]] || No || ||
|-
|[[Satisfies property::stem group]] || Yes || the center equals the derived subgroup, and hence, in particular, is contained in the derived subgroup. ||
|-
|[[Dissatisfies property::Schur-trivial group]] || No || the [[Schur multiplier]] is [[cyclic group:Z2]]; see [[group cohomology of dihedral group:D8]]. ||
|-
| [[dissatisfies property::A-group]] || No || The only Sylow subgroup is itself, which is not abelian. || Joint smallest non-A-group, alongside [[quaternion group]].
|-
|[[Satisfies property::Group isomorphic to its automorphism group]] || Yes || ||
|-
|[[Satisfies property::2-Engel group]] || Yes || ||
|}

==Subgroups==
{{further|[[subgroup structure of dihedral group:D8]]}}

[[File:D8latticeofsubgroups.png|800px|Lattice of subgroups of the dihedral group]]

{{#lst:subgroup structure of dihedral group:D8|summary}}

===Subgroup-defining functions and associated quotient-defining functions===

{{further|[[subgroup structure of dihedral group:D8#Defining functions]]}}
{{#lst:subgroup structure of dihedral group:D8|sdf summary}}

==Automorphisms and endomorphisms==

{{further|[[Endomorphism structure of dihedral group:D8]]}}
{{#lst:endomorphism structure of dihedral group:D8|summary}}

==Linear representation theory==

{{further|[[Linear representation theory of dihedral group:D8]], [[linear representation theory of dihedral groups]]}}

===Summary===

{{#lst:linear representation theory of dihedral group:D8|summary}}

===Character table===

{{#lst:linear representation theory of dihedral group:D8|character table}}

==Galois theory==

===Galois extensions===

{{further|[[Galois extensions for dihedral group:D8]]}}

==Fusion systems==

{{further|[[fusion systems for dihedral group:D8]]}}
===Summary===
{{#lst:fusion systems for dihedral group:D8|summary}}

===Description of fusion systems===
{{#lst:fusion systems for dihedral group:D8|description}}

==Distinguishing features==

===Smallest of its kind===

* This is the unique non-[[T-group]] of smallest order, i.e., the unique smallest example of a group in which [[normality is not transitive]].
* This is a non-abelian nilpotent group of smallest order, though not the only one. The other such group is the [[quaternion group]].

===Different from others of the same order===

* It is the only group of its order that is isomorphic to its [[automorphism group]].
* It is the only group of its order that is not a [[T-group]].
* It is the only group of its order having two Klein four-subgroups. In particular, it gives an example of a situation where the number of elementary abelian subgroups of order <math>p^2</math> is neither zero nor <math>1</math> modulo <math>p</math>. Contrast this with the case of odd <math>p</math>, where we have the [[congruence condition on number of elementary abelian subgroups of prime-square order for odd prime]].

==GAP implementation==
{{access GAP implementation online using SAGE|id=5013}}
{{GAP ID|8|3}}
{{HallSenior|8|4}}
===Short descriptions===

{| class="sortable" border="1"
! Description !! GAP functions used !! Mathematical translation of description
|-
| <tt>DihedralGroup(8)</tt> || [[GAP:DihedralGroup|DihedralGroup]] || dihedral group of order <math>8</math>, degree <math>4</math>
|-
| <tt>WreathProduct(CyclicGroup(2),CyclicGroup(2))</tt> || [[GAP:WreathProduct|WreathProduct]], [[GAP:CyclicGroup|CyclicGroup]] || [[external wreath product]] of two copies of cyclic group of order two
|-
| <tt>ExtraspecialGroup(2^3,'+')</tt> || [[GAP:ExtraspecialGroup|ExtraspecialGroup]] || [[extraspecial group]] of '+' type for the prime <math>2</math> and order <math>2^3</math>
|-
| <tt>SylowSubgroup(SymmetricGroup(4),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]] and [[GAP:SymmetricGroup|SymmetricGroup]] || The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S4|symmetric group of degree four]]
|-
| <tt>SylowSubgroup(GL(3,2),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]], [[GAP:GL|GL]] || The <math>2</math>-Sylow subgroup of [[GL(3,2)]]
|}

===Description by presentation===

Here is the code:

<pre>gap> F := FreeGroup(2);;
gap> G := F/[F.1^4, F.2^2, F.2 * F.1 * F.2 * F.1];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 8, 3 ]</pre>

The group <math>G</math> constructed here is the dihedral group of order <math>8</math>. The first generator <math>F.1</math> maps to the ''rotation'' element of order four and the second generator <math>F.2</math> maps to a ''reflection'' element of order two.

===Long descriptions===

It can be described as the [[holomorph of a group|holomorph]] of the cyclic group of order four. For this, first define <math>C</math> to be the cyclic group of order four (using [[GAP:CyclicGroup|CyclicGroup]]), and then use [[GAP:SemidirectProduct|SemidirectProduct]] and [[GAP:AutomorphismGroup|AutomorphismGroup]]:

<pre>C := CyclicGroup(4);
G := SemidirectProduct(AutomorphismGroup(C),C);</pre>

Here, <math>G</math> is the dihedral group of order eight. We can also construct it as a semidirect product of the Klein four-group and an automorphism of order two.

<pre>K := DirectProduct(CyclicGroup(2),CyclicGroup(2));
A := AutomorphismGroup(K);
S := SylowSubgroup(A,2);
G := SemidirectProduct(S,K);</pre>

Then, <math>G</math> is isomorphic to the dihedral group of order eight.

===GAP verification===

Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set <tt>G := DihedralGroup(8);</tt> or any equivalent way of setting <math>G</math> to be dihedral of order eight.

<pre>gap> IdGroup(G);
[ 8, 3 ]
gap> Order(G);
8
gap> Exponent(G);
4
gap> NilpotencyClassOfGroup(G);
2</pre>

More: <toggledisplay>

<pre>gap> DerivedLength(G);
2
gap> FrattiniLength(G);
2
gap> Rank(G);
2
gap> SubgroupRank(G);
2
gap> RankAsPGroup(G);
2
gap> NormalRank(G);
2
gap> CharacteristicRank(G);
1
gap> Length(ConjugacyClasses(G));
5
gap> Length(RationalClasses(G));
5
gap> Length(Subgroups(G));
10
gap> Length(ConjugacyClassesSubgroups(G));
8
gap> Length(NormalSubgroups(G));
6
gap> Length(CharacteristicSubgroups(G));
4</pre></toggledisplay>

Quaternion group

2024-12-03T00:35:16Z

R-a-jones: /* Other properties */

{{particular group}}
{{TOCright}}
[[importance rank::2| ]]
[[Category:Generalized quaternion groups]]
==Definition==

===Definition by presentation===

The quaternion group has the following [[presentation of a group|presentation]]:

<math>\langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle</math>

The identity is denoted <math>1</math>, the common element <math>i^2 = j^2 = k^2 = ijk</math> is denoted <math>-1</math>, and the elements <math>i^3, j^3, k^3</math> are denoted <math>-i,-j,-k</math> respectively.

{{quotation|Confused about presentations in general or this one in particular? If you're new to this stuff, check out [[constructing quaternion group from its presentation]]. Sophisticated group theorists can read [[equivalence of presentations of dicyclic group]]}}
===Verbal definitions===

The '''quaternion group''' is a group with eight elements, which can be described in any of the following ways:

* It is the group comprising eight elements <math>1,-1,i,-i,j,-j,k,-k</math> where 1 is the identity element, <math>(-1)^2 = 1</math> and all the other elements are squareroots of <math>-1</math>, such that <math>(-1)i = -i, (-1)j = -j, (-1)k= -k</math> and further, <math>ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j</math> (the remaining relations can be deduced from these).
* It is the {{dicyclic group}} with parameter 2, viz <math>Dic_2</math>.
* It is the [[member of family::Fibonacci group]] <math>F(2,3)</math>.

===Multiplication table===

{{#lst:element structure of quaternion group|multiplication table}}

This image shows visually the [[Cayley table]] of the quaternion group, with each element given a different colour. The elements are organized as in the above table: the first column/row corresponds to <math>1</math>, the second column/row corresponds to <math>-1</math>, etc.

[[File:Quaternion group Cayley table visual.png|200px]]

==Position in classifications==

{| class="sortable" border="1"
! Type of classification !! Name in that classification
|-
| GAP ID || (8,4), i.e., the 4th among the groups of order 8
|-
| Hall-Senior number || 5 among groups of order 8
|-
| Hall-Senior symbol || <math>8\Gamma_2a_2</math>
|}

==Elements==

{{further|[[Element structure of quaternion group]]}}

===Conjugacy class structure===
{{#lst:element structure of quaternion group|conjugacy and automorphism class structure}}

==Arithmetic functions==

===Basic arithmetic functions===

{{compare and contrast arithmetic functions|order = 8}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|8}} ||
|-
| {{arithmetic function value order p-log|3}} ||
|-
| {{arithmetic function value given order|exponent of a group|4|8}} || Cyclic subgroup of order four.
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|8|3}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|8|3}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|8|3}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|8|3}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|8|3}} || Generators of two cyclic subgroups of order four.
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|2|8|3}} || All proper subgroups are cyclic.
|-
| {{arithmetic function value given order and p-log|rank of a p-group|1|8|3}} || All abelian subgroups are cyclic.
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|1|8|3}} || All abelian normal subgroups are cyclic.
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|1|8|3}} || All abelian characteristic subgroups are cyclic.
|}

===Arithmetic functions of an element-counting nature===

{{further|[[element structure of quaternion group]]}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value given order|number of conjugacy classes|5|8}} || See [[element structure of dicyclic groups]].
|-
| {{arithmetic function value given order|number of equivalence classes under real conjugacy|5|8}} || Same as number of conjugacy classes, because the group is an [[ambivalent group]].
|-
| {{arithmetic function value given order|number of conjugacy classes of real elements|5|8}} || Same as number of conjugacy clases, because the group is an [[ambivalent group]].
|-
| {{arithmetic function value given order|number of equivalence classes under rational conjugacy|5|8}} || Same as number of conjugacy classes, because the group is a [[rational group]] (though not a [[rational representation group]]).
|-
| {{arithmetic function value given order|number of conjugacy classes of rational elements|5|8}} || Same as number of conjugacy classes, because the group is a [[rational group]] (though not a [[rational representation group]]).
|}
===Arithmetic functions of a subgroup-counting nature===

{{further|[[subgroup structure of quaternion group]]}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value|number of subgroups|6}} || ||
|-
| {{arithmetic function value|number of conjugacy classes of subgroups|6}} || ||
|-
| {{arithmetic function value given order|number of normal subgroups|6|8}} ||
|-
| {{arithmetic function value|number of automorphism classes of subgroups|4}} || ||
|}

===Lists of numerical invariants===

{| class="sortable" border="1"
! List !! Value !! Explanation/comment
|-
| [[conjugacy class size set|conjugacy class sizes]] || <math>1,1,2,2,2</math> || <math>\pm i, \pm j, \pm k</math> are each conjugacy classes of non-central elements.
|-
| [[degrees of irreducible representations]] || <math>1,1,1,1,2</math> || See [[linear representation theory of quaternion group]]
|-
| [[order statistics of a finite group|order statistics]] || <math>1 \mapsto 1, 2 \mapsto 1, 4 \mapsto 6</math> ||
|-
| orders of subgroups || <math>1,2,4,4,4,8</math> || See [[subgroup structure of quaternion group]]
|}

==Group properties==

{{compare and contrast group properties|order = 8}}

===Important properties===

{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
| {{group properties because p-group}}
|-
|[[Dissatisfies property::abelian group]] || No || <math>i</math> and <math>j</math> don't commute || Smallest non-abelian [[satisfies property::group of prime power order]]
|-
|[[Satisfies property::metacyclic group]] || Yes || Cyclic normal subgroup of order four, cyclic quotient of order two ||
|-
|[[Satisfies property::Dedekind group]] || Yes|| Every subgroup is normal || Smallest non-abelian Dedekind group
|-
|[[Satisfies property::T-group]] || Yes || Dedekind implies T-group ||
|}

===Other properties===

{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
|[[Satisfies property::monolithic group]] || Yes|| Unique minimal normal subgroup of order two ||
|-
|[[Dissatisfies property::one-headed group]] || No || Three distinct maximal normal subgroups of order four ||
|-
|[[Dissatisfies property::SC-group]] || No || ||
|-
|[[Satisfies property::ACIC-group]] || Yes || Every [[automorph-conjugate subgroup]] is [[characteristic subgroup|characteristic]] ||
|-
| [[Satisfies property::ambivalent group]] || Yes || ||
|-
|[[Satisfies property::rational group]] || Yes || Any two elements that generate the same cyclic group are conjugate || Thus, all characters are integer-valued.
|-
|[[Dissatisfies property::rational-representation group]] || No || [[Faithful irreducible representation of quaternion group|A two-dimensional representation that is not rational]]. || Contrast with [[dihedral group:D8]], that is rational-representation. See also [[linear representation theory of dihedral group:D8]] and [[linear representation theory of quaternion group]].
|-
| [[Satisfies property::maximal class group]] || Yes || ||
|-
| [[Satisfies property::group of nilpotency class two]] || Yes|| ||
|-
| [[Satisfies property::extraspecial group]] || Yes || ||
|-
| [[Satisfies property::special group]] || Yes || ||
|-
| [[Satisfies property::Frattini-in-center group]] || Yes || ||
|-
|[[Dissatisfies property::Frobenius group]] || No || Frobenius groups are centerless, and this group isn't. ||
|-
|[[Satisfies property::Camina group]] || Yes || [[extraspecial implies Camina]] ||
|-
|[[Satisfies property::group in which every element is automorphic to its inverse]] || Yes || Follows from being an [[ambivalent group]] ||
|-
|[[Satisfies property::group in which any two elements generating the same cyclic subgroup are automorphic]] || Yes || Follows from being a [[rational group]] ||
|-
|[[Satisfies property::group in which every element is order-automorphic]] || Yes || ||
|-
|[[Satisfies property::directly indecomposable group]] || Yes || ||
|-
|[[Satisfies property::centrally indecomposable group]] || Yes || ||
|-
|[[Satisfies property::splitting-simple group]] || Yes || ||
|-
|[[Satisfies property::Schur-trivial group]] || Yes || See [[group cohomology of quaternion group]] ||
|-
| [[dissatisfies property::A-group]] || No || The only Sylow subgroup is itself, which is not abelian. || Joint smallest non-A-group, alongside [[dihedral group:D8]].
|-
|[[Satisfies property::2-Engel group]] || Yes || ||
|}

==Subgroups==
{{further|[[Subgroup structure of quaternion group]]}}

[[Image:Q8latticeofsubgroups.png|500px]]
{{#lst:subgroup structure of quaternion group|summary}}

==Subgroup-defining functions and associated quotient-defining functions==

{{#lst:subgroup structure of quaternion group|sdf summary}}
==Automorphisms and endomorphisms==

{{further|[[endomorphism structure of quaternion group]]}}

{{#lst:endomorphism structure of quaternion group|summary}}

==Linear representation theory==

{{further|[[linear representation theory of quaternion group]]}}

===Summary===

{{#lst:linear representation theory of quaternion group|summary}}

===Character table===

{{#lst:linear representation theory of quaternion group|character table}}

==Distinguishing features==

===Smallest of its kind===

* This is a non-abelian [[nilpotent group]] of smallest possible order, along with [[dihedral group:D8]].
* This is a non-abelian [[Dedekind group]] (or Hamiltonian group) of smallest possible order. '''Dedekind''' means that every subgroup is normal.

===Different from others of the same order===

* It is the only non-abelian [[Dedekind group]] of its order.
* It is the only non-abelian [[T-group]] of its order.
* It is the only group of its order for which the [[rank of a p-group|rank]] (in the sense of the maximum possible rank of an abelian subgroup) is ''strictly'' smaller than the [[minimum size of generating set]]: For this group, the former is 1 and the latter is 2.
==GAP implementation==

{{GAP ID|8|4}}

{{HallSenior|8|5}}

===Short descriptions===

{| class="sortable" border="1"
! Description !! Functions used !! Mathematical comment
|-
| <tt>SylowSubgroup(SL(2,3),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]] and [[GAP:SL|SL]] || The <math>2</math>-Sylow subgroup of [[special linear group:SL(2,3)]]
|-
| <tt>ExtraspecialGroup(2^3,'-')</tt> || [[GAP:ExtraspecialGroup|ExtraspecialGroup]] || The extraspecial group of order <math>2^3</math> and '-' type
|-
| <tt>SylowSubgroup(SL(2,5),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]] and [[GAP:SL|SL]] || The <math>2</math>-Sylow subgroup of [[special linear group:SL(2,5)]]
|}

2-Engel group

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R-a-jones:

==Definition==

{| class="sortable" border="1"
! No. !! Shorthand !! A group is termed a Levi group or a 2-Engel group if ... !! A group <math>G</math> is termed a Levi group or 2-Engel group if ...
|-
| 1 || conjugates commute || any two [[conjugate elements]] of the group commute. || <math>x</math> commutes with <math>gxg^{-1}</math> for all <math>x,g \in G</math>
|-
| 2 || normal closures abelian || the [[defining ingredient::normal closure]] of any [[cyclic group|cyclic]] subgroup (or the [[defining ingredient::normal subgroup generated by a subset|normal subgroup generated]] by any one-element subset) is [[defining ingredient::abelian group|abelian]] || the normal closure of the subgroup generated by <math>x</math> is abelian for all <math>x \in G</math>.
|-
| 3 || union of abelian normal subgroups || the group is a ''union'' (as a set) of [[defining ingredient::abelian normal subgroup]]s || there is a collection of abelian normal subgroups <math>N_i, i \in I</math> of <math>G</math> such that <math>G = \bigcup_{i \in I} N_i</math>
|-
| 4 || 2-local class two || the 2-[[defining ingredient::local nilpotency class]] of the group is at most 2. || for any <math>x,g \in G</math>, the subgroup <math>\langle x,g \rangle</math> is a [[defining ingredient::group of nilpotency class two|group of class at most two]].
|-
| 5 || 2-Engel || the group is a <math>2</math>-[[defining ingredient::bounded Engel group|Engel group]]: the commutator between any element and its commutator with another element is the identity element. || the commutator <math>[x,[x,g]]</math> is the identity element for all <math>x,g \in G</math>.
|-
|6 || cyclic property of triple commutators || triple commutators are preserved under cyclic permutation of the inputs. ||for all <math>x,y,z \in G</math>, we have <math>[x,[y,z]] = [y,[z,x]] = [z,[x,y]]</math>.
|}

{{semistddef}}
{{group property}}

==Formalisms==

{{obtainedbyapplyingthe|Levi operator|Abelian group}}

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::abelian group]] || || || || {{intermediate notions short|Levi group|abelian group}}
|-
| [[Weaker than::Dedekind group]] || every subgroup is normal || || || {{intermediate notions short|Levi group|Dedekind group}}
|-
| [[Weaker than::group of nilpotency class two]] || [[nilpotency class]] at most two; or, quotient by [[center]] is an [[abelian group]] || || [[2-Engel not implies class two for groups]]|| {{intermediate notions short|Levi group|group of nilpotency class two}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::group generated by abelian normal subgroups]] || generated by abelian normal subgroups || || || {{intermediate notions short|group generated by abelian normal subgroups|Levi group}}
|-
| [[Stronger than::bounded Engel group]] || <math>k</math>-Engel group for some finite <math>k</math> || || || {{intermediate notions short|bounded Engel group|Levi group}}
|-
| [[Stronger than::Engel group]] || For any two elements <math>x</math> and <math>y</math>, the iterated commutator of <math>x</math> with <math>y</math> eventually becomes trivial || || || {{intermediate notions short|Engel group|Levi group}}
|-
| [[Stronger than::group in which order of commutator divides order of element]] || For any two elements <math>x</math> and <math>y</math>, if the order of <math>x</math> is finite, the order of <math>[x,y]</math> divides the order of <math>x</math> || || || {{intermediate notions short|group in which order of commutator divides order of element|Levi group}}
|-
| [[nilpotent group]] (for [[finite group]]s) || || || ||
|-
| [[Stronger than::Bell group]] || || || ||
|}

==Examples==

===Finite groups===

* Every finite [[abelian group]], since abelian groups are 2-Engel.
* The smallest non-abelian finite groups which are 2-Engel are both of the non-abelian groups of order 8: [[dihedral group:D8]] and [[quaternion group]]. Note in particular that [[symmetric group:S3]] is not 2-Engel.

2-Engel group

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R-a-jones:

Classification of groups of order a product of a prime-square and another prime

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R-a-jones: /* Statement */

==Statement==

Suppose <math>p</math> and <math>q</math> are ''distinct'' [[prime number]]s. This article classifies the groups of order <math>p^2q</math>.

===Case that <math>p</math> does not divide <math>q - 1</math> and <math>q</math> does not divide <math>p - 1</math>===

In this case, <math>p^2q</math> is an [[abelianness-forcing number]], i.e., all groups of this order are abelian. The two abelian groups are:

{| class="sortable" border="1"
! Group !! GAP ID second part !! Abelian? !! <math>p</math>-Sylow subgroup !! Is the <math>p</math>-Sylow subgroup normal? !! Is the <math>q</math>-Sylow subgroup normal?
|-
| cyclic group of order <math>p^2q</math> || 1 || Yes || [[cyclic group of prime-square order]] || Yes || Yes
|-
| direct product of cyclic group of order <math>p</math> and cyclic group of order <math>pq</math>, also same as direct product of elementary abelian group of order <math>p^2</math> and cyclic group of order <math>q</math> || 2 || Yes || [[elementary abelian group of prime-square order]] || Yes || Yes
|}

There are two examples of this below 100: the [[groups of order 45]] and the [[groups of order 99]].

===Case that <math>p</math> divides <math>q - 1</math> but <math>p^2</math> does not divide <math>q - 1</math> and <math>q</math> does not divide <math>p^2 - 1</math>===

In this case, there are four isomorphism classes of groups of order <math>p^2q</math>, given as follows:

{| class="sortable" border="1"
! Group !! GAP ID second part !! Abelian? !! <math>p</math>-Sylow subgroup !! Is the <math>p</math>-Sylow subgroup normal? !! Is the <math>q</math>-Sylow subgroup normal?
|-
| cyclic group of order <math>p^2q</math> || 1 || Yes || [[cyclic group of prime-square order]] || Yes || Yes
|-
| semidirect product of cyclic group of order <math>q</math> by cyclic group of order <math>p^2</math>, where the action by conjugation of a generator is an automorphism of order <math>p</math> || 2 || No || [[cyclic group of prime-square order]] || No || Yes
|-
| semidirect product of cyclic group of order <math>pq</math> by cyclic group of order <math>p</math>, where the action by conjugation of a generator is an automorphism of order <math>p</math> || 3 || No || [[elementary abelian group of prime-square order]] || No || Yes
|-
| direct product of cyclic group of order <math>p</math> and cyclic group of order <math>pq</math>, also same as direct product of elementary abelian group of order <math>p^2</math> and cyclic group of order <math>q</math> || 4 || Yes || [[elementary abelian group of prime-square order]] || Yes || Yes
|}

===Case that <math>p^2</math> divides <math>q - 1</math>===

In this case, there are five isomorphism classes of groups of order <math>p^2q</math>, given as follows:

{| class="sortable" border="1"
! Group !! GAP ID second part !! Abelian? !! <math>p</math>-Sylow subgroup !! Is the <math>p</math>-Sylow subgroup normal? !! Is the <math>q</math>-Sylow subgroup normal?
|-
| cyclic group of order <math>p^2q</math> || 1 || Yes || [[cyclic group of prime-square order]] || Yes || Yes
|-
| semidirect product of cyclic group of order <math>q</math> by cyclic group of order <math>p^2</math>, where the action by conjugation of a generator is an automorphism of order <math>p</math> || 2 || No || [[cyclic group of prime-square order]] || No || Yes
|-
| semidirect product of cyclic group of order <math>q</math> by cyclic group of order <math>p^2</math>, where the action by conjugation of a generator is an automorphism of order <math>p^2</math> || 3 || No || [[cyclic group of prime-square order]] || No || Yes
|-
| semidirect product of cyclic group of order <math>pq</math> by cyclic group of order <math>p</math>, where the action by conjugation of a generator is an automorphism of order <math>p</math> || 4 || No || [[elementary abelian group of prime-square order]] || No || Yes
|-
| direct product of cyclic group of order <math>p</math> and cyclic group of order <math>pq</math>, also same as direct product of elementary abelian group of order <math>p^2</math> and cyclic group of order <math>q</math> || 5 || Yes || [[elementary abelian group of prime-square order]] || Yes || Yes
|}

===Case that <math>q</math> divides <math>p - 1</math> but does not divide <math>p + 1</math>===

{{fillin}}

===Case that <math>q</math> divides <math>p + 1</math> but does not divide <math>p - 1</math>===

{{fillin}}

===Special cases: order 12 and order 18===

There are five groups of order 12. See [[classification of groups of order 12]]. The unusual example is [[alternating group:A4]].

There are five groups of order 18. See [[classification of groups of order 18]].

SmallGroup(28,4)

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R-a-jones: Redirected page to Direct product of Z2 and Z14

#redirect [[direct product of Z2 and Z14]]

SmallGroup(28,2)

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R-a-jones: Redirected page to Cyclic group:Z28

#redirect [[Cyclic group:Z28]]

SmallGroup(28,1)

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R-a-jones: Redirected page to Dicyclic group:Dic28

#redirect [[Dicyclic group:Dic28]]

SmallGroup(28,3)

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R-a-jones: Redirected page to Dihedral group:D28

#redirect [[Dihedral group:D28]]

SmallGroup(26,1)

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R-a-jones: Redirected page to Dihedral group:D26

#redirect [[Dihedral group:D26]]

SmallGroup(26,2)

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R-a-jones: Redirected page to Cyclic group:Z26

#redirect [[Cyclic group:Z26]]

SmallGroup(25,2)

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R-a-jones: Redirected page to Elementary abelian group:E25

#redirect [[Elementary abelian group:E25]]

SmallGroup(25,1)

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R-a-jones: Redirected page to Cyclic group:Z25

#redirect [[Cyclic group:Z25]]

Groups of order 25

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R-a-jones:

{{groups of order|25}}

There are, up to isomorphism, ''two'' possibilities for a group of order 25. Both of these are [[abelian group]]s and, in particular are [[abelian group of prime power order|abelian of prime power order]].

The classification follows from the [[classification of groups of prime-square order]].

See also [[groups of prime-square order]] for side-by-side comparison with the situation for other primes.

The groups are:

{| class="sortable" border="1"
! Group !! GAP ID (second part) !! Defining feature
|-
| [[cyclic group:Z25]] || 1 || unique [[cyclic group]] of order 25
|-
| [[elementary abelian group:E25]] || 2 || unique [[elementary abelian group]] of order 25; also a direct product of two copies of [[cyclic group:Z5]].
|}

SmallGroup(35,1)

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R-a-jones: Redirected page to Cyclic group:Z35

#redirect [[Cyclic group:Z35]]

SmallGroup(33,1)

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R-a-jones: Redirected page to Cyclic group:Z33

#redirect [[Cyclic group:Z33]]

SmallGroup(22,1)

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R-a-jones: Redirected page to Dihedral group:D22

#redirect [[Dihedral group:D22]]

SmallGroup(22,2)

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R-a-jones: Redirected page to Cyclic group:Z22

#redirect [[Cyclic group:Z22]]

SmallGroup(21,2)

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R-a-jones: Redirected page to Cyclic group:Z21

#redirect [[Cyclic group:Z21]]

User:R-a-jones/SmallGroup redirect pages

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[[SmallGroup(35,1)]]

[[SmallGroup(36,1)]]
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[[SmallGroup(64,267)]]

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[[SmallGroup(66,4)]]

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[[SmallGroup(68,4)]]
[[SmallGroup(68,5)]]

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[[SmallGroup(70,4)]]

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[[SmallGroup(72,47)]]
[[SmallGroup(72,48)]]
[[SmallGroup(72,49)]]
[[SmallGroup(72,50)]]

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[[SmallGroup(74,2)]]

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[[SmallGroup(75,2)]]
[[SmallGroup(75,3)]]

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[[SmallGroup(76,2)]]
[[SmallGroup(76,3)]]
[[SmallGroup(76,4)]]

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[[SmallGroup(78,2)]]
[[SmallGroup(78,3)]]
[[SmallGroup(78,4)]]
[[SmallGroup(78,5)]]
[[SmallGroup(78,6)]]

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[[SmallGroup(80,50)]]
[[SmallGroup(80,51)]]
[[SmallGroup(80,52)]]

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[[SmallGroup(81,2)]]
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[[SmallGroup(81,14)]]
[[SmallGroup(81,15)]]

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[[SmallGroup(82,2)]]

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[[SmallGroup(84,2)]]
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[[SmallGroup(84,12)]]
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[[SmallGroup(84,14)]]
[[SmallGroup(84,15)]]

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[[SmallGroup(86,1)]]
[[SmallGroup(86,2)]]

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[[SmallGroup(88,2)]]
[[SmallGroup(88,3)]]
[[SmallGroup(88,4)]]
[[SmallGroup(88,5)]]
[[SmallGroup(88,6)]]
[[SmallGroup(88,7)]]
[[SmallGroup(88,8)]]
[[SmallGroup(88,9)]]
[[SmallGroup(88,10)]]
[[SmallGroup(88,11)]]
[[SmallGroup(88,12)]]

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[[SmallGroup(90,2)]]
[[SmallGroup(90,3)]]
[[SmallGroup(90,4)]]
[[SmallGroup(90,5)]]
[[SmallGroup(90,6)]]
[[SmallGroup(90,7)]]
[[SmallGroup(90,8)]]
[[SmallGroup(90,9)]]
[[SmallGroup(90,10)]]

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[[SmallGroup(92,2)]]
[[SmallGroup(92,3)]]
[[SmallGroup(92,4)]]

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[[SmallGroup(93,2)]]

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[[SmallGroup(94,2)]]

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[[SmallGroup(96,90)]]
[[SmallGroup(96,91)]]
[[SmallGroup(96,92)]]
[[SmallGroup(96,93)]]
[[SmallGroup(96,94)]]
[[SmallGroup(96,95)]]
[[SmallGroup(96,96)]]
[[SmallGroup(96,97)]]
[[SmallGroup(96,98)]]
[[SmallGroup(96,99)]]
[[SmallGroup(96,100)]]
[[SmallGroup(96,101)]]
[[SmallGroup(96,102)]]
[[SmallGroup(96,103)]]
[[SmallGroup(96,104)]]
[[SmallGroup(96,105)]]
[[SmallGroup(96,106)]]
[[SmallGroup(96,107)]]
[[SmallGroup(96,108)]]
[[SmallGroup(96,109)]]
[[SmallGroup(96,110)]]
[[SmallGroup(96,111)]]
[[SmallGroup(96,112)]]
[[SmallGroup(96,113)]]
[[SmallGroup(96,114)]]
[[SmallGroup(96,115)]]
[[SmallGroup(96,116)]]
[[SmallGroup(96,117)]]
[[SmallGroup(96,118)]]
[[SmallGroup(96,119)]]
[[SmallGroup(96,120)]]
[[SmallGroup(96,121)]]
[[SmallGroup(96,122)]]
[[SmallGroup(96,123)]]
[[SmallGroup(96,124)]]
[[SmallGroup(96,125)]]
[[SmallGroup(96,126)]]
[[SmallGroup(96,127)]]
[[SmallGroup(96,128)]]
[[SmallGroup(96,129)]]
[[SmallGroup(96,130)]]
[[SmallGroup(96,131)]]
[[SmallGroup(96,132)]]
[[SmallGroup(96,133)]]
[[SmallGroup(96,134)]]
[[SmallGroup(96,135)]]
[[SmallGroup(96,136)]]
[[SmallGroup(96,137)]]
[[SmallGroup(96,138)]]
[[SmallGroup(96,139)]]
[[SmallGroup(96,140)]]
[[SmallGroup(96,141)]]
[[SmallGroup(96,142)]]
[[SmallGroup(96,143)]]
[[SmallGroup(96,144)]]
[[SmallGroup(96,145)]]
[[SmallGroup(96,146)]]
[[SmallGroup(96,147)]]
[[SmallGroup(96,148)]]
[[SmallGroup(96,149)]]
[[SmallGroup(96,150)]]
[[SmallGroup(96,151)]]
[[SmallGroup(96,152)]]
[[SmallGroup(96,153)]]
[[SmallGroup(96,154)]]
[[SmallGroup(96,155)]]
[[SmallGroup(96,156)]]
[[SmallGroup(96,157)]]
[[SmallGroup(96,158)]]
[[SmallGroup(96,159)]]
[[SmallGroup(96,160)]]
[[SmallGroup(96,161)]]
[[SmallGroup(96,162)]]
[[SmallGroup(96,163)]]
[[SmallGroup(96,164)]]
[[SmallGroup(96,165)]]
[[SmallGroup(96,166)]]
[[SmallGroup(96,167)]]
[[SmallGroup(96,168)]]
[[SmallGroup(96,169)]]
[[SmallGroup(96,170)]]
[[SmallGroup(96,171)]]
[[SmallGroup(96,172)]]
[[SmallGroup(96,173)]]
[[SmallGroup(96,174)]]
[[SmallGroup(96,175)]]
[[SmallGroup(96,176)]]
[[SmallGroup(96,177)]]
[[SmallGroup(96,178)]]
[[SmallGroup(96,179)]]
[[SmallGroup(96,180)]]
[[SmallGroup(96,181)]]
[[SmallGroup(96,182)]]
[[SmallGroup(96,183)]]
[[SmallGroup(96,184)]]
[[SmallGroup(96,185)]]
[[SmallGroup(96,186)]]
[[SmallGroup(96,187)]]
[[SmallGroup(96,188)]]
[[SmallGroup(96,189)]]
[[SmallGroup(96,190)]]
[[SmallGroup(96,191)]]
[[SmallGroup(96,192)]]
[[SmallGroup(96,193)]]
[[SmallGroup(96,194)]]
[[SmallGroup(96,195)]]
[[SmallGroup(96,196)]]
[[SmallGroup(96,197)]]
[[SmallGroup(96,198)]]
[[SmallGroup(96,199)]]
[[SmallGroup(96,200)]]
[[SmallGroup(96,201)]]
[[SmallGroup(96,202)]]
[[SmallGroup(96,203)]]
[[SmallGroup(96,204)]]
[[SmallGroup(96,205)]]
[[SmallGroup(96,206)]]
[[SmallGroup(96,207)]]
[[SmallGroup(96,208)]]
[[SmallGroup(96,209)]]
[[SmallGroup(96,210)]]
[[SmallGroup(96,211)]]
[[SmallGroup(96,212)]]
[[SmallGroup(96,213)]]
[[SmallGroup(96,214)]]
[[SmallGroup(96,215)]]
[[SmallGroup(96,216)]]
[[SmallGroup(96,217)]]
[[SmallGroup(96,218)]]
[[SmallGroup(96,219)]]
[[SmallGroup(96,220)]]
[[SmallGroup(96,221)]]
[[SmallGroup(96,222)]]
[[SmallGroup(96,223)]]
[[SmallGroup(96,224)]]
[[SmallGroup(96,225)]]
[[SmallGroup(96,226)]]
[[SmallGroup(96,227)]]
[[SmallGroup(96,228)]]
[[SmallGroup(96,229)]]
[[SmallGroup(96,230)]]
[[SmallGroup(96,231)]]

[[SmallGroup(97,1)]]

[[SmallGroup(98,1)]]
[[SmallGroup(98,2)]]
[[SmallGroup(98,3)]]
[[SmallGroup(98,4)]]
[[SmallGroup(98,5)]]

[[SmallGroup(99,1)]]
[[SmallGroup(99,2)]]

[[SmallGroup(100,1)]]
[[SmallGroup(100,2)]]
[[SmallGroup(100,3)]]
[[SmallGroup(100,4)]]
[[SmallGroup(100,5)]]
[[SmallGroup(100,6)]]
[[SmallGroup(100,7)]]
[[SmallGroup(100,8)]]
[[SmallGroup(100,9)]]
[[SmallGroup(100,10)]]
[[SmallGroup(100,11)]]
[[SmallGroup(100,12)]]
[[SmallGroup(100,13)]]
[[SmallGroup(100,14)]]
[[SmallGroup(100,15)]]
[[SmallGroup(100,16)]]

SmallGroup(15,1)

2024-08-26T17:13:02Z

R-a-jones: Redirected page to Cyclic group:Z15

#redirect [[cyclic group:Z15]]

SmallGroup(14,2)

2024-08-26T17:12:43Z

R-a-jones: Redirected page to Cyclic group:Z14

#redirect [[cyclic group:Z14]]

SmallGroup(14,1)

2024-08-26T17:12:32Z

R-a-jones: Redirected page to Dihedral group:D14

#redirect [[dihedral group:D14]]

Cyclic group:Z83

2024-08-26T17:09:31Z

R-a-jones: Created page with "{{particular group}} Category:Cyclic groups {{Cyclic group of prime order|83|<math>83</math>|<math>C_{83}, \Z_{83}, \Z / 83\Z</math>|eighty-three}}"

{{particular group}}
[[Category:Cyclic groups]]

{{Cyclic group of prime order|83|<math>83</math>|<math>C_{83}, \Z_{83}, \Z / 83\Z</math>|eighty-three}}

Cyclic group:Z79

2024-08-26T17:09:05Z

R-a-jones: Created page with "{{particular group}} Category:Cyclic groups {{Cyclic group of prime order|79|<math>79</math>|<math>C_{79}, \Z_{79}, \Z / 79\Z</math>|seventy-nine}}"

{{particular group}}
[[Category:Cyclic groups]]

{{Cyclic group of prime order|79|<math>79</math>|<math>C_{79}, \Z_{79}, \Z / 79\Z</math>|seventy-nine}}

Cyclic group:Z67

2024-08-26T17:08:34Z

R-a-jones: Created page with "{{particular group}} Category:Cyclic groups {{Cyclic group of prime order|67|<math>67</math>|<math>C_{67}, \Z_{67}, \Z / 67\Z</math>|sixty-seven}}"

{{particular group}}
[[Category:Cyclic groups]]

{{Cyclic group of prime order|67|<math>67</math>|<math>C_{67}, \Z_{67}, \Z / 67\Z</math>|sixty-seven}}

Cyclic group:Z53

2024-08-26T17:08:10Z

R-a-jones: Created page with "{{particular group}} Category:Cyclic groups {{Cyclic group of prime order|53|<math>53</math>|<math>C_{53}, \Z_{53}, \Z / 53\Z</math>|fifty-three}}"

{{particular group}}
[[Category:Cyclic groups]]

{{Cyclic group of prime order|53|<math>53</math>|<math>C_{53}, \Z_{53}, \Z / 53\Z</math>|fifty-three}}

SmallGroup(97,1)

2024-08-26T17:07:31Z

R-a-jones: Redirected page to Cyclic group:Z97

#redirect [[Cyclic group:Z97]]

SmallGroup(89,1)

2024-08-26T17:06:50Z

R-a-jones: Redirected page to Cyclic group:Z89

#redirect [[Cyclic group:Z89]]

SmallGroup(83,1)

2024-08-26T17:06:21Z

R-a-jones: Redirected page to Cyclic group:Z83

#redirect [[Cyclic group:Z83]]

SmallGroup(79,1)

2024-08-26T17:06:05Z

R-a-jones: Redirected page to Cyclic group:Z79

#redirect [[Cyclic group:Z79]]

SmallGroup(73,1)

2024-08-26T17:05:42Z

R-a-jones: Redirected page to Cyclic group:Z73

#redirect [[Cyclic group:Z73]]

SmallGroup(71,1)

2024-08-26T17:05:30Z

R-a-jones: Redirected page to Cyclic group:Z71

#redirect [[Cyclic group:Z71]]

SmallGroup(67,1)

2024-08-26T17:05:13Z

R-a-jones: Redirected page to Cyclic group:Z67

#redirect [[Cyclic group:Z67]]

SmallGroup(61,1)

2024-08-26T17:04:58Z

R-a-jones: Redirected page to Cyclic group:Z61

#redirect [[Cyclic group:Z61]]

SmallGroup(59,1)

2024-08-26T17:04:41Z

R-a-jones: Redirected page to Cyclic group:Z59

#redirect [[Cyclic group:Z59]]