Inner holomorph of a group
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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
.