Quaternion group

Quaternion group multiplication table (simplified form)
	$1$	$i$	$j$	$k$
$1$	$1$	$i$	$j$	$k$
$i$	$i$	$-1$	$k$	$- j$
$j$	$j$	$- k$	$-1$	$i$
$k$	$k$	$j$	$- i$	$-1$

In group theory, the quaternion group Q₈ (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset $\{1,i,j,k,-1,-i,-j,-k\}$ of the quaternions under multiplication. It is given by the group presentation

\mathrm {Q} _{8}=\langle {\bar {e}},i,j,k\mid {\bar {e}}^{2}=e,\;i^{2}=j^{2}=k^{2}=ijk={\bar {e}}\rangle ,

where e is the identity element and e commutes with the other elements of the group. These relations, discovered by W. R. Hamilton, also generate the quaternions as an algebra over the real numbers.

Another presentation of Q₈ is

\mathrm {Q} _{8}=\langle a,b\mid a^{4}=e,a^{2}=b^{2},ba=a^{-1}b\rangle .

Like many other finite groups, it can be realized as the Galois group of a certain field of algebraic numbers.