Quaternion group

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

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 , ). The assignment

defines a faithful representation of the quaternion group by complex -matrices.

A generalized quaternion group (a special case of which is the quaternion group for ) is a group defined on generators and and relations

(where is a fixed number). The group is a -group of order and nilpotency class .

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 -groups (cf. -group; Cyclic group) have this property. The generalized quaternion groups and the cyclic -groups are the only -groups admitting a proper -homomorphism, that is, a homomorphism of the lattice of subgroups onto some lattice 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 , of the quaternion group into the quaternion algebra gives a surjective algebra homomorphism of the group algebra to the quaternion algebra, exhibiting the latter as the quotient of by the ideal .