Holomorph of Z8

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This particular group is a finite group of order: 32

Definition

This group (which we shall call $G$ throughout) can be defined in either of these ways:

• It is the holomorph of the cyclic group on eight elements. In other words, it is the direct product of the cyclic group on eight elements, with its automorphism group.
• It is the holomorph of the ring $\mathbb{Z}/8\mathbb{Z}$.

Group properties

Solvability

This particular group is solvable

The group is solvable. In fact, it is metabelian, because the additive group is an Abelian normal subgroup (isomorphic to $\mathbb{Z}/8\mathbb{Z}$) and the quotient is Abelian, isomorphic to the Klein four-group.

The commutator subgroup of $G$ is not the whole of the additive group, though. It is only the even integers in the additive group.