# Quaternion group

A metabelian $2$-group (cf. Meta-Abelian group) of order 8, defined by generators $x,y$ and relations

$$x^4=x^2y^2=xyxy^{-1}=1.$$

The quaternion group can be isomorphically imbedded in the multiplicative group of the algebra of quaternions (cf. Quaternion; the imbedding is defined by the relation $x\mapsto i$, $y\mapsto j$). The assignment

$$x\mapsto\begin{pmatrix}0&1\\-1&0\end{pmatrix},y\mapsto\begin{pmatrix}0&i\\i&0\end{pmatrix}$$

defines a faithful representation of the quaternion group by complex $(2\times 2)$-matrices.

A generalized quaternion group (a special case of which is the quaternion group for $n=2$) is a group defined on generators $x$ and $y$ and relations

$$x^{2^n}=x^{2^{n-1}}y^2=xyxy^{-1}=1$$

(where $n$ is a fixed number). The group is a $2$-group of order $2^{n+1}$ and nilpotency class $n$.

The quaternion group is a Hamilton group, and the minimal Hamilton group in the sense that any non-Abelian Hamilton group contains a subgroup isomorphic to the quaternion group. The intersection of all non-trivial subgroups of the quaternion group (and also of any generalized quaternion group) is a non-trivial subgroup. Every non-Abelian finite group with this property is a generalized quaternion group. Among the finite Abelian groups, only the cyclic $p$-groups (cf. $p$-group; Cyclic group) have this property. The generalized quaternion groups and the cyclic $p$-groups are the only $p$-groups admitting a proper $L$-homomorphism, that is, a homomorphism of the lattice of subgroups onto some lattice $L$ that is not an isomorphism.

Sometimes the term "quaternion group" is used to denote various subgroups of the multiplicative group of the algebra of quaternions and related topological groups.

How to Cite This Entry:
Quaternion group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quaternion_group&oldid=32679
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article