Holomorph (mathematics)

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In mathematics, the holomorph of a group G is a semidirect product of G with its automorphism group Aut(G):

Hol(G) = G ⋊ Aut(G).

The semidirect product structure is the natural one, where the map

θ : Aut(G) → Aut(G)

is the identity. Therefore, more explicitly, Hol(G) may be given as the set of pairs (a, A) where a ∈ G and A ∈ Aut(G) with the composition rule

(a, A)(b, B) = (aA(b), AB).

There is a natural action of Hol(G) on G given by

(a, A)·x = aA(x)

for all x ∈ G. This action is faithful and transitive. It follows that Hol(G) may be embedded in the symmetric group on G. The image of Hol(G) in Sym(G) is generated by Aut(G) together with the group of left translations of G (see Cayley's theorem).

The construction of Hol(G) mirrors the construction of the affine group associated to a vector space. Indeed, Aff(V) is a subgroup of Hol(V), thinking of V as an abelian group. Moreover, for the finite vector space (C_p)ⁿ these two groups are actually equal (this is because every Z-linear automorphism of (C_p)ⁿ is also a C_p-linear automorphism).

It is easy to see that G, naturally embedded in Hol(G), is a normal subgroup of Hol(G):

(b, B)(a, 1)(b, B)⁻¹ = (bB(a)b⁻¹, 1).

That is, G is invariant under conjugation by elements of Hol(G). In particular, conjugation by (1,A) acts as the automorphism A on G. Phrased differently, inner automorphisms of Hol(G) when acting on G give the full automorphism group of G (i.e. the outer automorphisms of G come from inner automorphisms of Hol(G)).