Element structure of quaternion group

This article gives specific information, namely, element structure, about a particular group, namely: quaternion group.
View element structure of particular groups | View other specific information about quaternion group

This page discusses the element structure of the quaternion group. Notation for the quaternion group differs somewhat from notation for most groups. The multiplication table that we use throughout to identify elements is given below.

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

Element	$\!1$	$\!-1$	$\!i$	$\!-i$	$\!j$	$\!-j$	$\!k$	$\!-k$
$\!1$	$\!1$	$\!-1$	$\!i$	$\!-i$	$\!j$	$\!-j$	$\!k$	$\!-k$
$\!-1$	$\!-1$	$\!1$	$\!-i$	$\!i$	$\!-j$	$\!j$	$\!-k$	$\!k$
$\!i$	$\!i$	$\!-i$	$\!-1$	$\!1$	$\!k$	$\!-k$	$\!-j$	$\!j$
$\!-i$	$\!-i$	$\!i$	$\!1$	$\!-1$	$\!-k$	$\!k$	$\!j$	$\!-j$
$\!j$	$\!j$	$\!-j$	$\!-k$	$\!k$	$\!-1$	$\!1$	$\!i$	$\!-i$
$\!-j$	$\!-j$	$\!j$	$\!k$	$\!-k$	$\!1$	$\!-1$	$\!-i$	$\!i$
$\!k$	$\!k$	$\!-k$	$\!j$	$\!-j$	$\!-i$	$\!i$	$\!-1$	$\!1$
$\!-k$	$\!-k$	$\!k$	$\!-j$	$\!j$	$\!i$	$\!-i$	$\!1$	$\!-1$

Summary

Item	Value
order of the whole group (total number of elements)	8
conjugacy class sizes	1,1,2,2,2 maximum: 2, number of conjugacy classes: 5, lcm: 2
order statistics	1 of order 1, 1 of order 2, 6 of order 4 maximum: 4, lcm (exponent of the whole group): 4

Conjugacy and automorphism class structure

Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

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 $\langle i\rangle$
$\!\{j,-j\}$	2	4	$\{1,-1,j,-j\}$ -- same as $\langle j\rangle$
$\!\{k,-k\}$	2	4	$\{1,-1,k,-k\}$ -- same as $\langle k\rangle$

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

Order and power information

Directed power graph

Below is a trimmed version of the directed power graph of the group. There is an arrow from one vertex to another if the latter is the square of the former. We do not draw a loop at the identity element.

Order statistics

Number	Elements of order exactly that number	Number of such elements	Number of conjugacy classes of such elements	Number of elements whose order divides that number	Number of conjugacy classes whose element order divides that number
1	$\!\{1\}$	1	1	1	1
2	$\!\{-1\}$	1	1	2	2
4	$\|\{i,-i,j,-j,k,-k\}$	6	3	8	5

Power statistics

Number $d$	$d^{th}$ powers that are not $k^{th}$ powers for any larger divisor $k$ of the group order	Number of such elements	Number of conjugacy classes of such elements	Number of $d^{th}$ powers	Number of conjugacy classes of $d^{th}$ powers
1	$\!\{i,-i,j,-j,k,-k\}$	6	3	8	5
2	$\!\{-1\}$	1	1	2	2
4	--	0	0	1	1
8	$\!\{1\}$	1	1	1	1