Quaternion group

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

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

Dedekind group

The quaternion group is an example of a group in which every subgroup is normal, even though the group is not Abelian. In other words, it is a Dedekind group.

Nilpotence

This particular group is nilpotent

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.

Solvability

This particular group is solvable

The group is solvable of solvable length 2. The derived series is the same as the upper central series and lower central series.

Abelianness

This particular group is not Abelian

The group is not Abelian. In fact $i j \neq j i$ .

Simplicity

This particular group is not simple

The group is not simple. It has normal subgroups of order 2 and 4 (see below).

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.

Subgroup-defining functions

Center

The center of this group is abstractly isomorphic to: cyclic group of order two

The center of the quaternion group is the two-element subgroup comprising $- 1$ and $1$ .

Commutator subgroup

The commutator subgroup of this group is abstractly isomorphic to: cyclic group of order two

The commutator subgroup of the quaternion group is the same as its center: the two-element subgroup comprising $- 1$ and $1$ .

In particular this shows that the quaternion group is a group of nilpotence class two.

Frattini subgroup

The Frattini subgroup of this group is abstractly isomorphic to: 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

The socle of this group is abstractly isomorphic to: 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

The inner automorphism group of this group, viz the quotient group by its center, is abstractly isomorphic to: Klein four-group

When we quotient out by the center, every element has order two (because the square of every element is $\pm 1$ ). Hence the inner automorphism group is the Klein four-group. We can think of it as the quaternion group, modulo sign.

Abelianization

The Abelianization of this group, viz the quotient group by its commutator subgroup, is abstractly isomorphic to Klein four-group

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)

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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