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:

$\langle i,j,k\mid i^{2}=j^{2}=k^{2}=ijk\rangle$

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 $\mathbb {Z} /4\mathbb {Z}$ .
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, $ij=k,ji=-k,jk=i,kj=-1,ki=jik=-j$ (the remaining relations can be deduced from these).
It is the dicyclic group with parameter 2, viz $Dic_{2}$ .

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 $(p+1)^{th}$ .
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.

Arithmetic functions

Function	Value	Explanation
order	8
exponent	4	Cyclic subgroup of order four.
nilpotency class	2
derived length	2
Frattini length	2
Fitting length	1
minimum size of generating set	2	Generators of two cyclic subgroups of order four.
subgroup rank	2	All proper subgroups are cyclic.
max-length	3
rank as p-group	1	All abelian subgroups are cyclic.
normal rank	1	All abelian normal subgroups are cyclic.
characteristic rank of a p-group	1	All abelian characteristic subgroups are cyclic.

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

Further information: Subgroup structure of quaternion group

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	Page on subgroup embedding	Isomorphism class	Comment
Center	(2)	Center of quaternion group	Cyclic group:Z2	Prime power order implies not centerless
Commutator subgroup	(2)	Center of quaternion group	Cyclic group:Z2
Frattini subgroup	(2)	Center of quaternion group	Cyclic group:Z2	The three maximal subgroups of order four intersect here.
Socle	(2)	Center of quaternion group	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.
Frattini quotient	Klein four-group	It is the quotient by the Frattini subgroup, which is cyclic of order two.

Other associated constructs

Associated construct	Value (isomorphism class)	Comment
Automorphism group	symmetric group:S4
Outer automorphism group	symmetric group:S3

Supergroups

Further information: Supergroups of quaternion group

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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