Quaternion group

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 ${\displaystyle \{1,i,j,k,-1,-i,-j,-k\}}$ of the quaternions under multiplication. It is given by the group presentation

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

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

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

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