Editing Quaternion group (section)

==Generalized quaternion group==
A '''generalized quaternion group''' Q<sub>4''n''</sub> of order 4''n'' is defined by the presentation<ref name="Johnson44-45" />
:<math>\langle x,y \mid x^{2n} = y^4 = 1, x^n = y^2, y^{-1}xy = x^{-1}\rangle</math>
for an integer {{nowrap|''n'' ≥ 2}}, with the usual quaternion group given by ''n'' = 2.<ref>Some authors (e.g., {{harvnb|Rotman|1995}}, pp. 87, 351) refer to this group as the dicyclic group, reserving the name generalized quaternion group to the case where ''n'' is a power of 2.</ref> [[Harold Scott MacDonald Coxeter|Coxeter]] calls Q<sub>4''n''</sub> the [[dicyclic group]] <math>\langle 2, 2, n\rangle</math>, a special case of the [[binary polyhedral group]] <math>\langle \ell, m, n\rangle</math> and related to the [[polyhedral group]] <math>(p,q,r)</math> and the [[dihedral group]] <math>(2,2,n)</math>. The generalized quaternion group can be realized as the subgroup of <math>\operatorname{GL}_2(\Complex)</math> generated by

:<math>\left(\begin{array}{cc}
               \omega_n & 0 \\
               0 & \overline{\omega}_n
             \end{array}
          \right)
        \mbox{ and }
        \left(\begin{array}{cc}
                0 & -1 \\
                1 & 0
              \end{array}
          \right)
</math>

where <math>\omega_n = e^{i\pi/n}</math>.<ref name="Johnson44-45" /> It can also be realized as the subgroup of unit quaternions generated by<ref>{{harvnb|Brown|1982}}, p. 98</ref> <math>x=e^{i\pi/n}</math> and <math>y=j</math>.

The generalized quaternion groups have the property that every [[abelian group|abelian]] subgroup is cyclic.<ref>{{harvnb|Brown|1982}}, p. 101, exercise 1</ref> It can be shown that a finite [[p-group|''p''-group]] with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.<ref>{{harvnb|Cartan|Eilenberg|1999}}, Theorem 11.6, p. 262</ref> Another characterization is that a finite ''p''-group in which there is a unique subgroup of order ''p'' is either cyclic or a 2-group isomorphic to generalized quaternion group.<ref>{{harvnb|Brown|1982}}, Theorem 4.3, p. 99</ref> In particular, for a finite field ''F'' with odd characteristic, the 2-Sylow subgroup of SL<sub>2</sub>(''F'') is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, {{harv|Gorenstein|1980|p=42}}.  Letting ''p<sup>r</sup>'' be the size of ''F'', where ''p'' is prime, the size of the 2-Sylow subgroup of SL<sub>2</sub>(''F'') is 2<sup>''n''</sup>, where {{nowrap|1=''n'' = ord<sub>2</sub>(''p''<sup>2</sup> − 1) + ord<sub>2</sub>(''r'')}}.

The [[Brauer–Suzuki theorem]] shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.

Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2,<ref name="Roman">{{cite book|last1=Roman|first1=Steven|title=Fundamentals of Group Theory: An Advanced Approach|publisher=Springer|year=2011|isbn=9780817683016|pages=347–348|author-link1=Steven Roman}}</ref> which admits the presentation
:<math>\langle x,y \mid x^{2^m} = y^4 = 1, x^{2^{m-1}} = y^2, y^{-1}xy = x^{-1}\rangle.</math>
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-->==See also==
*[[16-cell]]
*[[Binary tetrahedral group]]
*[[Clifford algebra]]
*[[Dicyclic group]]
*[[Hurwitz integral quaternion]]
*[[List of small groups]]