Holomorph

The holomorph of a group G is the semidirect product

Hol(G) = G ⋊ Aut(G)

of G with its automorphism group, where Aut(G) acts on G tautologically — an automorphism α acts as the map α itself. Its order is

|Hol(G)| = |G| · |Aut(G)|.

It is the split extension 1 → G → Hol(G) → Aut(G) → 1 that packages a group together with all of its symmetries in one larger group.

Permutation realization

Inside the symmetric group Sym(G) of all permutations of the elements of G, take two subgroups:

• the right translations ρ(g): x ↦ x g, a copy of G acting regularly (Cayley's theorem);
• the automorphisms ᾱ: x ↦ α(x), a copy of Aut(G) fixing the identity e.

Conjugating a translation by an automorphism gives another translation,

ᾱ ∘ ρ(g) ∘ ᾱ⁻¹ = ρ(α(g)),

so the automorphisms normalize ρ(G); and the only automorphism that is also a translation is the identity, so the two copies meet trivially. Hence

Hol(G) = ⟨ρ(G), Ā⟩ = ρ(G) ⋊ Ā ≤ Sym(G).

In fact Hol(G) is exactly the normalizer NSym(G)(ρ(G)) — the largest group of permutations of G in which G's own translations form a normal subgroup.

Key facts

• Hol(G) contains G as a normal subgroup with complement Aut(G), and the extension always splits — unlike 1 → Inn(G) → Aut(G) → Out(G) → 1, which need not.
• Hol(G) is solvable ⟺ both G and Aut(G) are solvable (subgroups, quotients, and extensions of solvable groups are solvable).
• G complete (trivial center, no outer automorphisms) ⟹ Hol(G) ≅ G × G. (Then Aut(G) = Inn(G) ≅ G; the left and right translations λ(G), ρ(G) both lie in Hol(G), commute, meet trivially since Z(G) = 1, and have product of order |G|² = |Hol(G)|.)
• Hol(C₁) = C₁; and for a prime p, Hol(Cₚ) = AGL(1,𝔽p) has order p(p − 1) — a Frobenius group for every odd p (the complement Cp−1 acts fixed-point-freely), degenerating to just C₂ at p = 2.

Elementary abelian case: affine groups

When G = Cₚⁿ is elementary abelian, its automorphism group is the general linear group Aut(Cₚⁿ) = GL(n,𝔽p) acting linearly on the n-dimensional space over 𝔽ₚ, and the holomorph is the affine general linear group

Hol(Cₚⁿ) = 𝔽ₚⁿ ⋊ GL(n,𝔽p) = AGL(n,𝔽p),

the group of all invertible affine maps x ↦ Ax + b (see matrix groups). For example Hol(C₂²) = AGL(2,𝔽2) ≅ S₄, and Hol(C₂³) = AGL(3,𝔽2) has order 8 · 168 = 1344.

Examples

Hol(C₄) = D₄ shows that the holomorph of a cyclic group need not be affine: C₄ is cyclic but not elementary abelian, so Aut(C₄) = C₂ (inversion), not a GL. Holomorphs also leave the range of specially named groups quickly — even Hol(Q₈), with Aut(Q₈) ≅ S₄, is a group of order 8 · 24 = 192 with no standard name.

See also

• Automorphism group — the acting factor Aut(G).
• Semidirect product, Extension — Hol(G) is a split extension of Aut(G) by G.
• Matrix groups — Hol(Cₚⁿ) = AGL(n,𝔽p).
• Cayley's theorem — the regular representation ρ(G) ≤ Sym(G).
• Wikipedia: Holomorph