quaternion group

The quaternion group, or quaternionic group, is a noncommutative group with eight elements. It is traditionally denoted by $Q$ (not to be confused with $\mathbb{Q}$) or by $Q_8$. This group is defined by the presentation

$$\{i,j;i^4,i^2{j}^2,ij{i}^{-1}j\}$$

or, equivalently, defined by the multiplication table

$\cdot$	$1$	$i$	$j$	$k$	$-i$	$-j$	$-k$	$-1$
$1$	$1$	$i$	$j$	$k$	$-i$	$-j$	$-k$	$-1$
$i$	$i$	$-1$	$k$	$-j$	$1$	$-k$	$j$	$-i$
$j$	$j$	$-k$	$-1$	$i$	$k$	$1$	$-i$	$-j$
$k$	$k$	$j$	$-i$	$-1$	$-j$	$i$	$1$	$-k$
$-i$	$-i$	$1$	$-k$	$j$	$-1$	$k$	$-j$	$i$
$-j$	$-j$	$k$	$1$	$-i$	$-k$	$-1$	$i$	$j$
$-k$	$-k$	$-j$	$i$	$1$	$j$	$-i$	$-1$	$k$
$-1$	$-1$	$-i$	$-j$	$-k$	$i$	$j$	$k$	$1$

where we have put each product $xy$ into row $x$ and column $y$. The minus signs are justified by the fact that $\{1,-1\}$ is subgroup contained in the center of $Q$. Every subgroup of $Q$ is normal and, except for the trivial subgroup $\{1\}$, contains $\{1,-1\}$. The dihedral group $D_4$ (the group of symmetries of a square) is the only other noncommutative group of order 8.

Since $i^2=j^2=k^2=-1$, the elements $i$, $j$, and $k$ are known as the imaginary units, by analogy with $i\in ℂ$. Any pair of the imaginary units generate the group. Better, given $x,y\in \{i,j,k\}$, any element of $Q$ is expressible in the form $x^m{y}^n$.

$Q$ is identified with the group of units (invertible elements) of the ring of quaternions over $\mathbb{Z}$. That ring is not identical to the group ring $\mathbb{Z}[Q]$, which has dimension 8 (not 4) over $\mathbb{Z}$. Likewise the usual quaternion algebra is not quite the same thing as the group algebra $ℝ[Q]$.

Quaternions were known to Gauss in 1819 or 1820, but he did not publicize this discovery, and quaternions weren’t rediscovered until 1843, with Hamilton.

Title	quaternion group
Canonical name	QuaternionGroup
Date of creation	2013-03-22 12:35:35
Last modified on	2013-03-22 12:35:35
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	12
Author	mathcam (2727)
Entry type	Definition
Classification	msc 20A99
Synonym	quaternionic group
Related topic	Quaternions
Defines	quaternion group