Editing Quaternion group (section)

{{Short description|Non-abelian group of order eight}}
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{{Group theory sidebar |Finite}}
[[Image:GroupDiagramQ8.svg|240px|thumb|[[Cycle graph (group)|Cycle diagram]] of Q<sub>8</sub>. Each color specifies a series of powers of any element connected to the identity element e = 1. For example, the cycle in red reflects the fact that i<sup>2</sup> = {{overline|e}}, i<sup>3</sup> = {{overline|i}} and i<sup>4</sup> = e. The red cycle also reflects that {{overline|i}}<sup>2</sup> = {{overline|e}}, {{overline|i}}<sup>3</sup> = i and {{overline|i}}<sup>4</sup> = e.]]

In [[group theory]],  the '''quaternion group''' Q<sub>8</sub> (sometimes just denoted by Q) is a [[nonabelian group|non-abelian]] [[group (mathematics)|group]] of [[Group order|order]] eight, isomorphic to the eight-element subset
<math>\{1,i,j,k,-1,-i,-j,-k\}</math> of the [[quaternion]]s under multiplication. It is given by the [[presentation of a group|group presentation]]

:<math>\mathrm{Q}_8 = \langle \bar{e},i,j,k \mid \bar{e}^2 = e, \;i^2 = j^2 = k^2 = ijk = \bar{e} \rangle ,</math>

where ''e'' is the identity element and {{overline|''e''}} [[commutativity|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 Q<sub>8</sub> is
:<math>\mathrm{Q}_8 = \langle a,b \mid a^4 = e, a^2 = b^2, ba = a^{-1}b\rangle.</math>

Like many other finite groups, it [[Inverse Galois problem|can be realized]] as the [[#Galois group|Galois group]] of a certain field of [[algebraic number]]s.<ref name=":0" />