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

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.