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.

References

[1]	M. Hall jr., "Group theory" , Macmillan (1959)

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 $(x^2+1)$.