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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 $$\{i,j;i^4,i^2j^2,iji^{-1}j\}$$ 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 $i\in\mathbb{C}$ . 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 $x^my^n$ .
$Q$ 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 $\mathbb{R}[Q]$ .
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.