# Quaternion group

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## Definition

### Definition by presentation

The quaternion group has the following presentation:

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

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

### 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, $ij = k, ji = -k, jk = i, kj = -1, ki = j ik = -j$ (the remaining relations can be deduced from these).
• It is the dicyclic group with parameter 2, viz $Dic_2$.
• It is the Fibonacci group $F(2,3)$.

### 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

1. 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 $(p+1)^{th}$.
2. 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).
3. 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.

## Position in classifications

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 $8\Gamma_2a_2$

## Elements

### Upto conjugacy

The quaternion group has five conjugacy classes:

1. The identity element: This has order 1 and size 1
2. The element $-1$: This has order 2 and size 1
3. The two-element conjugacy class comprising $\pm i$: This has order 4 and size 2
4. The two-element conjugacy class comprising $\pm j$: This has order 4 and size 2
5. 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

Want to compare with other groups of the same order? Check out groups of order 8#Arithmetic functions.
Function Value Explanation
underlying prime of p-group 2
order 8
prime-base logarithm of order 3
exponent 4 Cyclic subgroup of order four.
prime-base logarithm of exponent 2
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.

### Lists of numerical invariants

List Value Explanation/comment
conjugacy class sizes $1,1,2,2,2$ $\pm i, \pm j, \pm k$ are each conjugacy classes of non-central elements.
degrees of irreducible representations $1,1,1,1,2$ See linear representation theory of quaternion group
order statistics $1 \mapsto 1, 2 \mapsto 1, 4 \mapsto 6$
orders of subgroups $1,2,4,4,4,8$ See subgroup structure of quaternion group

## Group properties

Want to compare with other groups of the same order? Check out groups of order 8#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
ambivalent group Yes
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.
maximal class group Yes
group of nilpotency class two Yes
extraspecial group Yes
special group Yes
Frattini-in-center group Yes
Frobenius group No Frobenius groups are centerless, and this group isn't.
Camina group Yes extraspecial implies Camina
group in which every element is automorphic to its inverse Yes Follows from being an ambivalent group
group in which any two elements generating the same cyclic subgroup are automorphic Yes Follows from being a rational group
group in which every element is order-automorphic Yes
directly indecomposable group Yes
centrally indecomposable group Yes
splitting-simple group Yes

## Subgroups

Further information: Subgroup structure of quaternion group

Lattice of subgroups of the quaternion group

The quaternion group has six subgroups:

1. The trivial subgroup (1)
2. The center, which is the unique minimal subgroup. This is a two-element subgroup comprising $\pm 1$ (1)
3. The three cyclic subgroups of order four, generated by $i, j, k$ respectively. These are all normal, but are automorphs of each other (3)
4. 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
Inner holomorph inner holomorph of D8 The inner holomorphs of $D_8$ and the quaternion group are isomorphic.

## Supergroups

Further information: Supergroups of quaternion group

## Implementation in GAP

### Group ID

This finite group has order 8 and has ID 4 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(8,4)

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

gap> G := SmallGroup(8,4);

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

IdGroup(G) = [8,4]

or just do:

IdGroup(G)

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