Quaternion group

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Definition

Definition by presentation

The quaternion group has the following presentation:

$⟨i,j,k\mid i^2=j^2=k^2=ijk⟩$

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 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, $ij=k,ji=-k,jk=i,kj=-i,ki=j,ik=-j$ (the remaining relations can be deduced from these).
• It is the dicyclic group with parameter 2, viz $Di{c}_2$.
• It is the Fibonacci group $F(2,3)$.

Multiplication table

In the table below, the row element is multiplied on the left and the column element on the right.

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

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	$8{\mathrm{\Gamma }}_2{a}_2$

Elements

Further information: Element structure of quaternion group

Conjugacy class structure

Conjugacy class	Size of conjugacy class	Order of elements in conjugacy class	Centralizer of first element of class
$\{1\}$	1	1	whole group
$\{-1\}$	1	2	whole group
$\{i,-i\}$	2	4	$\{1,-1,i,-i\}$, same as $⟨i⟩$
$\{j,-j\}$	2	4	$\{1,-1,j,-j\}$ -- same as $⟨j⟩$
$\{k,-k\}$	2	4	$\{1,-1,k,-k\}$ -- same as $⟨k⟩$

Automorphism class structure

Equivalence class (orbit) under action of automorphisms	Size of equivalence class (orbit)	Number of conjugacy classes in it	Size of each conjugacy class	Order of elements
$\{1\}$	1	1	1	1
$\{-1\}$	1	1	1	2
$\{i,-i,j,-j,k,-k\}$	6	3	2	4

Arithmetic functions

Basic arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions

Function	Value	Similar groups	Explanation
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	8	groups with same order
prime-base logarithm of order	3	groups with same prime-base logarithm of order
exponent of a group	4	groups with same order and exponent of a group | groups with same exponent of a group	Cyclic subgroup of order four.
prime-base logarithm of exponent	2	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	2	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
Frattini length	2	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	Generators of two cyclic subgroups of order four.
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.
rank of a p-group	1	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	All abelian subgroups are cyclic.
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.
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