nLab holomorph

This entry is about the notion in group theory. For holomorphic functions see there.

 Idea

In group theory, by the “holomorph” $Hol(G)$ of a (discrete) group $G$ one means the group equivalently described in any of the following ways:

• the smallest group $Hol(G)$ containing $G$ as a subgroup $\iota \colon G \hookrightarrow Hol(G)$ such that every automorphism of $G$ appears as an inner automorphism of $Hol(G)$ restricted along $\iota$,

• the normalizer $N_{Sym(G)}(G)$ of $G$ regarded as a subgroup (via its regular representation) of the symmetric group $Sym(G)$ on (i.e. the automorphism group of) its underlying set,

• the semidirect product group $G \rtimes Aut(G)$ of $G$ with its automorphism group (via the defining group action of $Aut(G)$ on $G$),

• the group $Mor\big(Aut(\mathbf{B} G)\big)$ of morphisms in the automorphism 2-group of $G$, regarded as the strict 2-group corresponding to the crossed module $G \xrightarrow{ad} Aut(G)$, where “$ad$” is the adjoint action of $G$ on itself (i.e. by inner automorphisms).

References

• Marshall Hall, §6.3 in: The Theory of Groups, Macmillan (1959), AMS Chelsea (1976), Dover (2018) [ISBN:978-0-8218-1967-8, ISBN:978-0-4868-1690-6]

• Wikipedia, Holomorph (mathematics)

• Groupprops, Holomorph of a group