Quaternion group: Difference between revisions

Revision as of 22:17, 13 May 2009

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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

This particular group is the smallest (in terms of order): nilpotent non-Abelian group

This particular group is the smallest (in terms of order): non-Abelian semidirectly indecomposable group

This particular group is the smallest (in terms of order): non-Abelian Dedekind group

This particular group is a finite group of order: 8

Definition

Definition by presentation

The quaternion group has the following presentation:

$⟨ i, j, k ∣ i^{2} = j^{2} = k^{2} = i j k ⟩$

Verbal definitions

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

It is the holomorph of the ring $Z / 4 Z$ .
It is the holomorph of the cyclic group of order 4.
It is the group comprising eight elements $1, - 1, i, - i, j, - j, k, - k$ where 1 is the identity element, $(- 1)^{2} = 1$ and all the other elements are squareroots of $- 1$ , such that $(- 1) i = - i, (- 1) j = - j, (- 1) k = - k$ and further, $i j = k, j i = - k, j k = i, k j = - 1, k i = j i k = - j$ (the remaining relations can be deduced from these).
It is the dicyclic group with parameter 2, viz $D i c_{2}$ .

Multiplication table

Element	$1$	$- 1$	$i$	$- i$	$j$	$- j$	$k$	$- k$
$1$	$1$	$- 1$	$i$	$- i$	$j$	$- j$	$k$	$- k$
$- 1$	$- 1$	$1$	$- i$	$i$	$- j$	$j$	$- k$	$k$
$i$	$i$	$- i$	$- 1$	$1$	$k$	$- k$	$- j$	$j$
$- i$	$- i$	$i$	$1$	$- 1$	$- k$	$k$	$j$	$- j$
$j$	$j$	$- j$	$- k$	$k$	$- 1$	$1$	$i$	$- i$
$- j$	$- j$	$j$	$k$	$- k$	$1$	$- 1$	$- i$	$i$
$k$	$k$	$- k$	$j$	$- j$	$- i$	$i$	$- 1$	$1$
$- k$	$- k$	$k$	$- j$	$j$	$i$	$- i$	$1$	$- 1$

Families

The construction of the quaternion group can be mimicked for other primes giving, in general, a non-Abelian group of order $p^{3}$ . The general construction involves taking a semidirect product of the cyclic group of order $p^{2}$ with a subgroup of order $p$ in the automorphism group, say the subgroup generated by the automorphism taking an element to its $(p + 1)^{t h}$ .
The quaternion group also generalizes to the family of dicyclic groups (also known as binary dihedral groups) and also to the family of generalized quaternion groups (which are the dicyclic groups whose order is a power of 2).
The quaternion group is part of a larger family of $p$ -groups called extraspecial groups. An extraspecial group is a group of prime power order whose center, commutator subgroup and Frattini subgroup coincide, and are all cyclic of prime order.

Elements

Upto conjugacy

The quaternion group has five conjugacy classes:

The identity element: This has order 1 and size 1
The element $- 1$ : This has order 2 and size 1
The two-element conjugacy class comprising $\pm i$ : This has order 4 and size 2
The two-element conjugacy class comprising $\pm j$ : This has order 4 and size 2
The two-element conjugacy class comprising $\pm k$ : This has order 4 and size 2

Upto automorphism

Under the action of automorphisms, the last three conjugacy classes get merged, so there are three equivalence classes, of sizes 1, 1, and 6.

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	No	$i$ and $j$ don't commute	Smallest non-abelian group of prime power order
Nilpotent group	Yes	Prime power order implies nilpotent	Smallest nilpotent non-abelian group, along with dihedral group:D8.
Metacyclic group	Yes	Cyclic normal subgroup of order four, cyclic quotient of order two
Supersolvable group	Yes	Metacyclic implies supersolvable
Solvable group	Yes	Metacyclic implies solvable
Dedekind group	Yes	Every subgroup is normal	Smallest non-abelian Dedekind group
T-group	Yes	Dedekind implies T-group
Monolithic group	Yes	Unique minimal normal subgroup of order two
One-headed group	No	Three distinct maximal normal subgroups of order four
SC-group	No
ACIC-group	Yes	Every automorph-conjugate subgroup is characteristic
Rational group	Yes	Any two elements that generate the same cyclic group are conjugate	Thus, all characters are integer-valued.
Rational-representation group	Yes	A two-dimensional representation that is not rational.	Contrast with dihedral group:D8, that is rational-representation.

Subgroups

The quaternion group has six subgroups:

The trivial subgroup (1)
The center, which is the unique minimal subgroup. This is a two-element subgroup comprising $\pm 1$ (1)
The three cyclic subgroups of order four, generated by $i, j, k$ respectively. These are all normal, but are automorphs of each other (3)
The whole group (1)

Normal subgroups

All subgroups are normal. The subgroups are the whole group, the trivial subgroup, the center, and three copies of the cyclic group on 4 elements. This makes the quaternion group a Dedekind group.

Characteristic subgroups

There are only three characteristic subgroups: the whole group, the trivial subgroup and the center.

Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class	Comment
Center	(2)	Cyclic group:Z2	Prime power order implies not centerless
Commutator subgroup	(2)	Cyclic group:Z2
Frattini subgroup	(2)	Cyclic group:Z2	The three maximal subgroups of order four intersect here.
Socle	(2)	Cyclic group:Z2	This subgroup is the unique minimal normal subgroup, i.e.,the monolith, and the group is monolithic. Also, minimal normal implies central in nilpotent.

=Quotient-defining functions

Quotient-defining function	Isomorphism class	Comment
Inner automorphism group	Klein four-group	It is the quotient by the center, which is of order two.
Abelianization	Klein four-group	It is the quotient by the commutator subgroup, which is cyclic of order two.

Implementation in GAP

Group ID

The quaternion group is the fourth group of order 8 in GAP, and can be described as:

SmallGroup(8,4)

Other descriptions

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Internal links

@@ Line 46: / Line 46: @@
 | <math>-k</math> || <math>-k</math> || <math>k</math> || <math>-j</math> || <math>j</math> || <math>i</math> || <math>-i</math> || <math>1</math> || <math>-1</math>
 |}
+==Families==
+# The construction of the quaternion group can  be mimicked for other primes giving, in general, a non-Abelian group of order <math>p^3</math>. The general construction involves taking a semidirect product of the cyclic group of order <math>p^2</math> with a subgroup of order <math>p</math> in the automorphism group, say the subgroup generated by the automorphism taking an element to its <math>(p+1)^{th}</math>.
+# The quaternion group also generalizes to the family of [[dicyclic group]]s (also known as binary dihedral groups) and also to the family of [[generalized quaternion group]]s (which are the dicyclic groups whose order is a power of 2).
+# The quaternion group is part of a larger family of <math>p</math>-groups called [[extraspecial group]]s. An extraspecial group is a group of prime power order whose center, commutator subgroup and Frattini subgroup coincide, and are all cyclic of prime order.
 ==Elements==
@@ Line 65: / Line 71: @@
 ==Group properties==
-===Dedekind group===
+{| class="wikitable" border="1"
-[[Category:Dedekind groups]]
+!Property !! Satisfied !! Explanation !! Comment
+|-
-The quaternion group is an example of a group in which ''every'' subgroup is [[normal subgroup|normal]], even though the group is not [[Abelian group|Abelian]]. In other words, it is a [[Dedekind group]].
+|[[Dissatisfies property::Abelian group]] || No || <math>i</math> and <math>j</math> don't commute || Smallest non-abelian group of prime power order
+|-
-{{nilpotent}}
+|[[Satisfies property::Nilpotent group]] || Yes || [[Prime power order implies nilpotent]] || Smallest nilpotent non-abelian group, along with [[dihedral group:D8]].
+|-
-In fact, the [[upper central series]] and [[lower central series]] of the quaternion group are both of length two, and comprise the same three members: the trivial subgroup, the two-element subgroup which is the center (1 and -1) and the whole group.
+|[[Satisfies property::Metacyclic group]] || Yes || Cyclic normal subgroup of order four, cyclic quotient of order two ||
+|-
-{{solvable}}
+|[[Satisfies property::Supersolvable group]] || Yes || [[Metacyclic implies supersolvable]] ||
+|-
-The group is solvable of solvable length 2. The [[derived series]] is the same as the upper central series and lower central series.
+|[[Satisfies property::Solvable group]] || Yes || Metacyclic implies solvable ||
+|-
-{{not Abelian}}
+|[[Satisfies property::Dedekind group]] || Yes|| Every subgroup is normal || Smallest non-abelian Dedekind group
+|-
-The group is not Abelian. In fact <math>ij \ne ji</math>.
+|[[Satisfies property::T-group]] || Yes || Dedekind implies T-group ||
+|-
-{{not simple}}
+|[[Satisfies property::Monolithic group]] || Yes|| Unique minimal normal subgroup of order two ||
+|-
-The group is not simple. It has normal subgroups of order 2 and 4 (see below).
+|[[Dissatisfies property::One-headed group]] || No || Three distinct maximal normal subgroups of order four ||
+|-
-==Families==
+|[[Dissatisfies property::SC-group]] || No ||  ||
+|-
-# The construction of the quaternion group can  be mimicked for other primes giving, in general, a non-Abelian group of order <math>p^3</math>. The general construction involves taking a semidirect product of the cyclic group of order <math>p^2</math> with a subgroup of order <math>p</math> in the automorphism group, say the subgroup generated by the automorphism taking an element to its <math>(p+1)^{th}</math>.
+|[[Satisfies property::ACIC-group]] || Yes || Every [[automorph-conjugate subgroup]] is [[characteristic subgroup|characteristic]] ||
-# The quaternion group also generalizes to the family of [[dicyclic group]]s (also known as binary dihedral groups) and also to the family of [[generalized quaternion group]]s (which are the dicyclic groups whose order is a power of 2).
+|-
-# The quaternion group is part of a larger family of <math>p</math>-groups called [[extraspecial group]]s. An extraspecial group is a group of prime power order whose center, commutator subgroup and Frattini subgroup coincide, and are all cyclic of prime order.
+|[[Satisfies property::Rational group]] || Yes || Any two elements that generate the same cyclic group are conjugate || Thus, all characters are integer-valued.
+|-
-==Subgroup-defining functions==
+|[[Dissatisfies property::Rational-representation group]] || Yes || A two-dimensional representation that is not rational. || Contrast with [[dihedral group:D8]], that is rational-representation.
+|}
-{{center|cyclic group of order two}}
-The center of the quaternion group is the two-element subgroup comprising <math>-1</math> and <math>1</math>.
-{{commutator subgroup|cyclic group of order two}}
-The commutator subgroup of the quaternion group is the same as its center: the two-element subgroup comprising <math>-1</math> and <math>1</math>.
-In particular this shows that the quaternion group is a group of nilpotence class two.
-{{frattini subgroup|cyclic group of order two}}
-The Frattini subgroup is also the same as the center and commutator subgroup. In fact, this makes the quaternion group into an [[extraspecial group]].
-{{socle|cyclic group of order two}}
-The center is the unique minimal normal subgroup, and hence also functions as the socle.
-==Quotient-defining functions==
-{{inner automorphism group|[[Klein four-group]]}}
-When we quotient out by the center, every element has order two (because the square of every element is <math>\pm 1</math>). Hence the inner automorphism group is the Klein four-group. We can think of it as the quaternion group, modulo sign.
-{{Abelianization|[[Klein four-group]]}}
 ==Subgroups==
@@ Line 138: / Line 119: @@
 There are only three characteristic subgroups: the whole group, the trivial subgroup and the center.
-{{fully characteristic subgroups same as characteristic}}
+==Subgroup-defining functions==
+{| class="wikitable" border="1"
+! Subgroup-defining function !! Subgroup type in list !! Isomorphism class !! Comment
+|-
+| [[Center]] || (2) || [[Center::Cyclic group:Z2]] || [[Prime power order implies not centerless]]
+|-
+| [[Commutator subgroup]] || (2) || [[Commutator subgroup::Cyclic group:Z2]] ||
+|-
+| [[Frattini subgroup]] || (2) || [[Frattini subgroup::Cyclic group:Z2]] || The three maximal subgroups of order four intersect here.
+|-
+| [[Socle]] || (2) || [[Socle::Cyclic group:Z2]] || This subgroup is the unique [[minimal normal subgroup]], i.e.,the [[monolith]], and the group is [[monolithic group|monolithic]]. Also, [[minimal normal implies central in nilpotent]].
+|}
+===Quotient-defining functions==
+{| class="wikitable" border="1"
+! Quotient-defining function !! Isomorphism class !! Comment
+|-
+| [[Inner automorphism group]] || [[Inner automorphism group::Klein four-group]] || It is the quotient by the center, which is of order two.
+|-
+| [[Abelianization]] || [[Abelianization::Klein four-group]] || It is the quotient by the commutator subgroup, which is cyclic of order two.
+|}
 ==Implementation in GAP==