Inner holomorph of a group

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Definition

Let G be a group. The inner holomorph of G can be defined as the semidirect product G \rtimes \operatorname{Inn}(G) where \operatorname{Inn}(G) is the inner automorphism group with the usual action.

It is a subgroup of the holomorph G \rtimes \operatorname{Aut}(G) and is a quotient of the direct product G \times G.

Facts

When G is an group having an automorphism whose restriction to the center is the inverse map, this is isomorphic to the central product of two copies of G with the center Z(G) of both copies identified: G *_{Z(G)} G.

If G is a group whose center is a direct factor, this group is isomorphic to the direct product of G and \operatorname{Inn}(G).