quaternion group

The quaternion group, or quaternionic group, is a noncommutativegroup with eight elements. It is traditionally denoted by $Q$ (not to beconfused with $\mathbb{Q}$) or by $Q_8$. This group is defined by thepresentation

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

or, equivalently, defined by the multiplication table

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 subgroupcontained in the center of $Q$.Every subgroup of $Q$ is normal and, except forthe trivial subgroup $\{1\}$, contains $\{1,-1\}$.The dihedral group $D_4$ (the group of symmetries of a square) is theonly 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, byanalogy with $i\in ℂ$. Any pair of the imaginary units generatethe 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 thering of quaternions over $\mathbb{Z}$. That ringis not identical to the group ring $\mathbb{Z}[Q]$, which has dimension 8(not 4) over $\mathbb{Z}$. Likewise the usual quaternion algebrais not quite the same thing as the group algebra $ℝ[Q]$.

Quaternions were known to Gauss in 1819 or 1820, but he did notpublicize this discovery, and quaternions weren’t rediscovered until1843, with Hamilton.