holomorph in nLab

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

Contents

 Idea

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

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

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

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

• the group Mor(Aut(BG))Mor\big(Aut(\mathbf{B} G)\big) of morphisms in the automorphism 2-group of GG, regarded as the strict 2-group corresponding to the crossed module G→adAut(G)G \xrightarrow{ad} Aut(G), where “adad” is the adjoint action of GG 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]

• A. G. Kurosh, Teorija grupp (Теория групп), 2 vols., Nauka 1944, 2nd edition 1953. English transl. The theory of groups, Chelsea, NY 1960 archive

• Wikipedia, Holomorph (mathematics)

• Groupprops, Holomorph of a group