View source for Quaternion group

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A metabelian $2$-group (cf. [[Meta-Abelian group|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|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|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|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|$p$-group]]; [[Cyclic 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hall jr.,   "Group theory" , Macmillan  (1959)</TD></TR></table>



====Comments====
The imbedding $x\mapsto i$, $y\mapsto j$ of the quaternion group $H$ into the quaternion algebra gives a surjective algebra homomorphism of the group algebra $\mathbf R[H]$ to the quaternion algebra, exhibiting the latter as the quotient of $\mathbf R[H]$ by the ideal $(x^2+1)$.