# 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

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

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 $x^{m}y^{n}$.

$Q$ 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 $\mathbb{R}[Q]$.

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.

Title quaternion group QuaternionGroup 2013-03-22 12:35:35 2013-03-22 12:35:35 mathcam (2727) mathcam (2727) 12 mathcam (2727) Definition msc 20A99 quaternionic group Quaternions quaternion group