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).

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]
A. G. Kurosh (translated by K. Hirsch); §13 in: Theory of Groups, Chelsea Publishing (1960, 1965) [ark:/13960/s201vm968q1, pdf]