Semidihedral group:SD16: Difference between revisions

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The semidihedral group $S D_{16}$ (also denoted $Q D_{16}$ ) is the semidihedral group (also called quasidihedral group) of order 16. Specifically, it has the following presentation:

$S D_{16} : = ⟨ a, x ∣ a^{8} = x^{2} = e, x a x^{- 1} = a^{3} ⟩$ .

The group can also be defined as the general semilinear group of degree one over the field of nine elements.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	groups with same order and exponent of a group \| groups with same exponent of a group	cyclic subgroup of order 8.
prime-base logarithm of exponent	3	groups with same order and prime-base logarithm of exponent \| groups with same prime-base logarithm of order and prime-base logarithm of exponent \| groups with same prime-base logarithm of exponent
nilpotency class	3	groups with same order and nilpotency class \| groups with same prime-base logarithm of order and nilpotency class \| groups with same nilpotency class
derived length	2	groups with same order and derived length \| groups with same prime-base logarithm of order and derived length \| groups with same derived length	the derived subgroup is contained in the cyclic subgroup and is hence abelian
Frattini length	3	groups with same order and Frattini length \| groups with same prime-base logarithm of order and Frattini length \| groups with same Frattini length
minimum size of generating set	2	groups with same order and minimum size of generating set \| groups with same prime-base logarithm of order and minimum size of generating set \| groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group \| groups with same prime-base logarithm of order and subgroup rank of a group \| groups with same subgroup rank of a group	All proper subgroups are cyclic, dihedral, semidihedral or Klein four-groups.
rank of a p-group	2	groups with same order and rank of a p-group \| groups with same prime-base logarithm of order and rank of a p-group \| groups with same rank of a p-group	there exist Klein four-subgroups.
normal rank of a p-group	1	groups with same order and normal rank of a p-group \| groups with same prime-base logarithm of order and normal rank of a p-group \| groups with same normal rank of a p-group	all abelian normal subgroups are cyclic.
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group \| groups with same prime-base logarithm of order and characteristic rank of a p-group \| groups with same characteristic rank of a p-group	All abelian characteristic subgroups are cyclic.

Note that the order can also be computed using the formula for the order of $Γ L (1, p^{r})$ where $p = 3, r = 2$ and $q = p^{r} = 9$ : The order of $Γ L (1, q)$ with $q = p^{r}$ is $r (q - 1)$ , which becomes $2 (9 - 1) = 2 (8) = 16$ .

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation
number of conjugacy classes	7	groups with same order and number of conjugacy classes \| groups with same prime-base logarithm of order and number of conjugacy classes \| groups with same number of conjugacy classes	As $S D_{2^{n}}, n = 4$ : $2^{n - 2} + 3 = 2^{2} + 3 = 7$ As $Γ L (1, p^{2}), p = 3$ : $(p^{2} + 3 p - 4) / 2$ which becomes $(3^{2} + 3 \cdot 3 - 4) / 2 = 7$ . See element structure of general semilinear group of degree one over a finite field. See element structure of semidihedral group:SD16
number of equivalence classes under rational conjugacy	6	groups with same order and number of equivalence classes under rational conjugacy \| groups with same prime-base logarithm of order and number of equivalence classes under rational conjugacy \| groups with same number of equivalence classes under rational conjugacy
number of conjugacy classes of rational elements	5	groups with same order and number of conjugacy classes of rational elements \| groups with same prime-base logarithm of order and number of conjugacy classes of rational elements \| groups with same number of conjugacy classes of rational elements

Group properties

Important properties

Property	Satisfied	Explanation
group of prime power order	Yes
nilpotent group	Yes	prime power order implies nilpotent
supersolvable group	Yes	via nilpotent: finite nilpotent implies supersolvable
solvable group	Yes	via nilpotent: nilpotent implies solvable
abelian group	No
metacyclic group	Yes	has a cyclic subgroup of order 8, with cyclic quotient group
metabelian group	Yes	follows from being metacyclic; also, from having class three.

Other properties

Property	Meaning	Satisfied?	Explanation	Comment
group in which every normal subgroup is characteristic	every normal subgroup of the group is a characteristic subgroup of it.	Yes
maximal class group	finite p-group that is non-abelian and whose nilpotency class is one less than its prime-base logarithm of order.	Yes
UL-equivalent group	nilpotent group whose upper central series and lower central series coincide.	Yes	follows from being a maximal class group
stem group	center is contained in the derived subgroup.	Yes
directly indecomposable group	nontrivial and cannot be expressed as an internal direct product of proper nontrivial subgroups.	Yes
centrally indecomposable group	nontrivial and cannot be expressed as an internal central product of proper nontrivial subgroups.	Yes
splitting-simple group	nontrivial and cannot be expressed as an internal semidirect product of proper nontrivial subgroups.	No	it is an internal semidirect product of the normal subgroup $⟨ a ⟩$ and the group $⟨ x ⟩$ .
finite group with periodic cohomology	finite group in which every abelian subgroup is cyclic. For the finite p-group case, this is equivalent to the rank being one.	No
Schur-trivial group	the Schur multiplier is trivial.	Yes
ambivalent group	every element is conjugate to its inverse.	No	$a, a^{- 1}$ are not conjugate.
ACIC-group	every automorph-conjugate subgroup is characteristic	No	$⟨ a^{4}, x ⟩$ is automorph-conjugate but not characteristic.
T-group	every subnormal subgroup is normal	No	$⟨ x ⟩$ is subnormal but not normal.	Also, not a Dedekind group. In fact, for nilpotent groups, being a T-group is equivalent to being a Dedekind group.

Subgroups

Further information: Subgroup structure of semidihedral group:SD16

Automorphism class of subgroups	List of subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if subgroup is normal)	Subnormal depth (if proper and normal, this equals 1)	Nilpotency class
trivial subgroup	${e}$	trivial group	1	16	1	1	1	semidihedral group:SD16	1	0
center of semidihedral group:SD16	${e, a^{4}}$	cyclic group:Z2	2	8	1	1	1	dihedral group:D8	1	1
non-normal order two subgroups of semidihedral group:SD16	${e, x}, {e, a^{2} x}, {e, a^{4} x}, {e, a^{6} x}$	cyclic group:Z2	2	8	1	4	4	--	3	1
derived subgroup of semidihedral group:SD16	${e, a^{2}, a^{4}, a^{6}}$	cyclic group:Z4	4	4	1	1	1	Klein four-group	1	1
non-normal order four cyclic subgroups of semidihedral group:SD16	${e, a x, a^{4}, a^{5} x}$ , ${e, a^{3} x, a^{4}, a^{7} x}$	cyclic group:Z4	4	4	1	2	2	--	2	1
Klein four-subgroups of semidihedral group:SD16	${e, x, a^{4}, a^{4} x}$ , ${e, a^{2} x, a^{4}, a^{6} x}$	Klein four-group	4	4	1	2	2	--	2	1
Z8 in SD16	$⟨ a ⟩$	cyclic group:Z8	8	2	1	1	1	cyclic group:Z2	1	1
D8 in SD16	$⟨ a^{2}, x ⟩$	dihedral group:D8	8	2	1	1	1	cyclic group:Z2	1	2
Q8 in SD16	$⟨ a^{2}, a x ⟩$	quaternion group	8	2	1	1	1	cyclic group:Z2	1	2
whole group	all elements	semidihedral group:SD16	16	1	1	1	1	trivial group	0	3
Total (10 rows)	--	--	--	--	10	--	15	--	--	--

There are a couple of interesting facts about this group:

Every subgroup of this group is an automorph-conjugate subgroup.
All the maximal subgroups are characteristic subgroups -- in fact, they are all isomorph-free subgroups.

Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class
Center	(2)	cyclic group:Z2
Commutator subgroup	(4)	cyclic group:Z4
Frattini subgroup	(4)	cyclic group:Z4
Socle	(2)	cyclic group:Z4
Join of abelian subgroups of maximum order	(7)	cyclic group:Z8
Join of abelian subgroups of maximum rank	(8)	dihedral group:D8
Join of elementary abelian subgroups of maximum order	(8)	dihedral group:D8

GAP implementation

Group ID

This finite group has order 16 and has ID 8 among the groups of order 16 in GAP's SmallGroup library. For context, there are groups of order 16. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(16,8)

For instance, we can use the following assignment in GAP to create the group and name it $G$ :

gap> G := SmallGroup(16,8);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [16,8]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Hall-Senior number

This group of prime power order has order 16 and has Hall-Senior number 13 among the groups of order 16. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

Gap3CatalogueGroup(16,13)

WARNING: There is some disagreement between the GAP 3 catalogue numbers and the Hall-Senior numbers for some abelian groups, but it does not affect this group.

For instance, we can use the following assignment in GAP to create the group and name it $G$ :

gap> G := Gap3CatalogueGroup(16,13);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's Gap3CatalogueIdGroup function:

Gap3CatalogueIdGroup(G) = [16,13]

or just do:

Gap3CatalogueIdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Description by presentation

The group can be defined using a presentation as follows:

gap> F := FreeGroup(2);;
gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-3)];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 16, 8 ]

Other descriptions

Description	Functions used
`SylowSubgroup(GL(2,3),2)`	SylowSubgroup, GL

@@ Line 1: / Line 1: @@
 {{particular group}}
+[[importance rank::3| ]]
 ==Definition==
-The semidihedral group <math>SD_{16}</math> is the [[semidihedral group]] of order <math>16</math>. Specifically, it has the following presentation:
+The semidihedral group <math>SD_{16}</math> (also denoted <math>QD_{16}</math>)  is the [[semidihedral group]] (also called '''quasidihedral group''') of [[order 16]]. Specifically, it has the following presentation:
 <math>SD_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \rangle</math>.
+The group can also be defined as the [[general semilinear group]] of [[general semilinear group of degree one|degree one]] over [[field:F9|the field of nine elements]].
+==Arithmetic functions==
+{{semidihedral 2-group arithmetic function table|
+order = 16|
+order p-log = 4|
+degree = 8|
+degree p-log = 3}}
+Note that the order can also be computed using the formula for the order of <math>\Gamma L(1,p^r)</math> where <math>p = 3, r = 2</math> and <math>q = p^r = 9</math>: The order of <math>\Gamma L(1,q)</math> with <math>q = p^r</math> is <math>r(q - 1)</math>, which becomes <math>2(9 - 1) = 2(8) = 16</math>.
+===Arithmetic functions of a counting nature===
+{| class="sortable" border="1"
+! Function !! Value !! Similar groups !! Explanation
+|-
+| {{arithmetic function value given order and p-log|number of conjugacy classes|7|16|4}} || As <math>SD_{2^n}, n = 4</math>: <math>2^{n-2} + 3 = 2^2 + 3 = 7</math><br>As <math>\Gamma L(1,p^2), p = 3</math>: <math>(p^2 + 3p - 4)/2</math> which becomes <math>(3^2 + 3 \cdot 3 - 4)/2 = 7</math>. See [[element structure of general semilinear group of degree one over a finite field]].<br>See [[element structure of semidihedral group:SD16]]
+|-
+| {{arithmetic function value given order and p-log|number of equivalence classes under rational conjugacy|6|16|4}} ||
+|-
+| {{arithmetic function value given order and p-log|number of conjugacy classes of rational elements|5|16|4}} ||
+|}
+==Group properties==
+===Important properties===
+{| class="sortable" border="1"
+! Property !! Satisfied !! Explanation !! Comment
+|-
+| {{group properties because p-group}}
+|-
+| [[dissatisfies property::abelian group]] || No || ||
+|-
+| [[satisfies property::metacyclic group]] || Yes || has a cyclic subgroup of order 8, with cyclic quotient group ||
+|-
+| [[satisfies property::metabelian group]] || Yes || follows from being metacyclic; also, from having class three. ||
+|}
+===Other properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Satisfied? !! Explanation !! Comment
+|-
+| [[satisfies property::group in which every normal subgroup is characteristic]] || every [[normal subgroup]] of the group is a [[characteristic subgroup]] of it. || Yes || ||
+|-
+| [[satisfies property::maximal class group]] || [[finite p-group]] that is non-abelian and whose nilpotency class is one less than its [[prime-base logarithm of order]]. || Yes || ||
+|-
+| [[satisfies property::UL-equivalent group]] || [[nilpotent group]] whose [[upper central series]] and [[lower central series]] coincide. || Yes || follows from being a [[maximal class group]] ||
+|-
+| [[satisfies property::stem group]] || [[center]] is contained in the [[derived subgroup]]. || Yes || ||
+|-
+| [[satisfies property::directly indecomposable group]] || nontrivial and cannot be expressed as an [[internal direct product]] of proper nontrivial subgroups. || Yes || ||
+|-
+| [[satisfies property::centrally indecomposable group]] || nontrivial and cannot be expressed as an [[internal central product]] of proper nontrivial subgroups. || Yes || ||
+|-
+| [[dissatisfies property::splitting-simple group]] || nontrivial and cannot be expressed as an [[internal semidirect product]] of proper nontrivial subgroups. || No || it is an internal semidirect product of the normal subgroup <math>\langle a \rangle</math> and the group <math>\langle x \rangle</math>. ||
+|-
+| [[dissatisfies property::finite group with periodic cohomology]] || finite group in which every abelian subgroup is cyclic. For the [[finite p-group]] case, this is equivalent to the [[rank of a p-group|rank]] being one. || No || ||
+|-
+| [[satisfies property::Schur-trivial group]] || the [[Schur multiplier]] is trivial. || Yes || ||
+|-
+| [[dissatisfies property::ambivalent group]] || every element is conjugate to its inverse. || No || <math>a,a^{-1}</math> are not conjugate. ||
+|-
+| [[dissatisfies property::ACIC-group]] || every [[automorph-conjugate subgroup]] is [[characteristic subgroup|characteristic]] || No || <math>\langle a^4,x \rangle</math> is automorph-conjugate but not characteristic. ||
+|-
+| [[dissatisfies property::T-group]] || every [[subnormal subgroup]] is [[normal subgroup|normal]] || No || <math>\langle x \rangle</math> is subnormal but not normal. || Also, not a [[dissatisfies property::Dedekind group]]. In fact, for nilpotent groups, being a T-group is equivalent to being a Dedekind group.
+|}
 ==Subgroups==
@@ Line 11: / Line 80: @@
 {{further|[[Subgroup structure of semidihedral group:SD16]]}}
-Here is a list of subgroups.
+{{#lst:subgroup structure of semidihedral group:SD16|summary}}
-# The trivial subgroup. Isomorphic to [[subgroup::trivial group]]. (1).
-# The subgroup of order two generated by <math>a^4</math>. This is the [[center]] of the whole group. Isomorphic to [[subgroup::cyclic group:Z2]]. (1)
-# Subgroups of order two generated by the elements <math>x, a^2x, a^4x, a^6x</math>. These are all [[conjugate subgroups]]. Isomorphic to [[subgroup::cyclic group:Z2]]. (4)
-# Subgroups of order four generated by the elements <math>ax, a^3x</math>. These are [[conjugate subgroups]]. Note that <math>\langle ax\rangle</math> contains <math>e, ax, a^4, a^5x</math>. Isomorphic to [[subgroup::cyclic group:Z4]]. (2)
-# Subgroup of order four generated by the element <math>a^2</math>. This is a [[characteristic subgroup]] and is the [[commutator subgroup]] of the whole group. Isomorphic to [[subgroup::cyclic group:Z4]]. (1)
-# Subgroups of order four: <math>\langle a^4,x \rangle, \langle a^4,a^2x \rangle</math>. These are both [[conjugate subgroups]]. Isomorphic to [[subgroup::Klein four-group]]. (2)
-# Cyclic subgroup <math>\langle a \rangle</math> of order eight. Isomorphic to [[subgroup::cyclic group:Z8]]. (1)
-# Subgroup <math>\langle a^2, x \rangle</math> of order eight. Isomorphic to [[subgroup::dihedral group:D8]]. (1)
-# Subgroup <math>\langle a^2, ax \rangle</math> of order eight. Isomorphic to [[subgroup::quaternion group]]. (1)
-# The whole group. (1)
 There are a couple of interesting facts about this group:
@@ Line 28: / Line 86: @@
 * Every subgroup of this group is an [[automorph-conjugate subgroup]].
 * All the maximal subgroups are [[characteristic subgroup]]s -- in fact, they are all [[isomorph-free subgroup]]s.
-==Quotient groups==
-# The [[quotient group::semidihedral group:SD16| ]] group itself. Obtained as the quotient by the trivial subgroup. (1)
-# The quotient by the [[center]], or the [[inner automorphism group]]. Isomorphic to [[quotient group::dihedral group:D8]]. (1)
-# The quotient by the [[commutator subgroup]], or the [[abelianization]]. Isomorphic to [[quotient group::Klein four-group]]. (1)
-# The quotient by the cyclic subgroup of order eight. Isomorphic to [[quotient group::cyclic group:Z2]]. (1)
-# The quotient by the dihedral subgroup of order eight. Isomorphic to [[quotient group::cyclic group:Z2]]. (1)
-# The quotient group by the quaternion group. Isomorphic to [[quotient group::cyclic group:Z2]]. (1)
-# The [[trivial group]]. (1)
 ==Subgroup-defining functions==
@@ Line 61: / Line 109: @@
 {{GAP ID|16|8}}
+{{HallSenior|16|13}}
+===Description by presentation===
+The group can be defined using a presentation as follows:
+<pre>gap> F := FreeGroup(2);;
+gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-3)];
+<fp group on the generators [ f1, f2 ]>
+gap> IdGroup(G);
+[ 16, 8 ]</pre>
+===Other descriptions===
+{| class="sortable" border="1"
+! Description !! Functions used
+|-
+| <tt>SylowSubgroup(GL(2,3),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]], [[GAP:GL|GL]]
+|}