# Inner holomorph of a group

## Definition

Let be a group. The **inner holomorph** of can be defined as the semidirect product where is the inner automorphism group with the usual action.

It is a subgroup of the holomorph and is a quotient of the direct product .

## Facts

When 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 with the center of both copies identified: .

If is a group whose center is a direct factor, this group is isomorphic to the direct product of and .