Holomorph (mathematics)

Hol(G) as a semi-direct product

If Aut ⁡ ( G ) is the automorphism group of G then

Hol ⁡ ( G ) = G ⋊ Aut ⁡ ( G )

where the multiplication is given by

( g , α ) ( h , β ) = ( g α ( h ) , α β ) . [Eq. 1]

Typically, a semidirect product is given in the form G ⋊ ϕ A where G and A are groups and ϕ : A → Aut ⁡ ( G ) is a homomorphism and where the multiplication of elements in the semi-direct product is given as

( g , a ) ( h , b ) = ( g ϕ ( a ) ( h ) , a b )

which is well defined, since ϕ ( a ) ∈ Aut ⁡ ( G ) and therefore ϕ ( a ) ( h ) ∈ G .

For the holomorph, A = Aut ⁡ ( G ) and ϕ is the identity map, as such we suppress writing ϕ explicitly in the multiplication given in [Eq. 1] above.

For example,

G = C 3 = ⟨ x ⟩ = { 1 , x , x 2 } the cyclic group of order 3

Aut ⁡ ( G ) = ⟨ σ ⟩ = { 1 , σ } where σ ( x ) = x 2

Hol ⁡ ( G ) = { ( x i , σ j ) } with the multiplication given by:

( x i 1 , σ j 1 ) ( x i 2 , σ j 2 ) = ( x i 1 + i 2 2 j 1 , σ j 1 + j 2 ) where the exponents of x are taken mod 3 and those of σ mod 2.

Observe, for example

( x , σ ) ( x 2 , σ ) = ( x 1 + 2 ⋅ 2 , σ 2 ) = ( x 2 , 1 )

and note also that this group is not abelian, as ( x 2 , σ ) ( x , σ ) = ( x , 1 ) , so that Hol ⁡ ( C 3 ) is a non-abelian group of order 6 which, by basic group theory, must be isomorphic to the symmetric group S 3 .

Hol(G) as a permutation group

A group G acts naturally on itself by left and right multiplication, each giving rise to a homomorphism from G into the symmetric group on the underlying set of G. One homomorphism is defined as λ: G → Sym(G), λ(g)(h) = g·h. That is, g is mapped to the permutation obtained by left multiplying each element of G by g. Similarly, a second homomorphism ρ: G → Sym(G) is defined by ρ(g)(h) = h·g⁻¹, where the inverse ensures that ρ(g·h)(k) = ρ(g)(ρ(h)(k)). These homomorphisms are called the left and right regular representations of G. Each homomorphism is injective, a fact referred to as Cayley's theorem.

For example, if G = C₃ = {1, x, x² } is a cyclic group of order three, then

λ(x)(1) = x·1 = x,

λ(x)(x) = x·x = x², and

λ(x)(x²) = x·x² = 1,

so λ(x) takes (1, x, x²) to (x, x², 1).

The image of λ is a subgroup of Sym(G) isomorphic to G, and its normalizer in Sym(G) is defined to be the holomorph N of G. For each n in N and g in G, there is an h in G such that n·λ(g) = λ(h)·n. If an element n of the holomorph fixes the identity of G, then for 1 in G, (n·λ(g))(1) = (λ(h)·n)(1), but the left hand side is n(g), and the right side is h. In other words, if n in N fixes the identity of G, then for every g in G, n·λ(g) = λ(n(g))·n. If g, h are elements of G, and n is an element of N fixing the identity of G, then applying this equality twice to n·λ(g)·λ(h) and once to the (equivalent) expression n·λ(g·h) gives that n(g)·n(h) = n(g·h). That is, every element of N that fixes the identity of G is in fact an automorphism of G. Such an n normalizes λ(G), and the only λ(g) that fixes the identity is λ(1). Setting A to be the stabilizer (group theory) of the identity, the subgroup generated by A and λ(G) is semidirect product with normal subgroup λ(G) and complement A. Since λ(G) is transitive, the subgroup generated by λ(G) and the point stabilizer A is all of N, which shows the holomorph as a permutation group is isomorphic to the holomorph as semidirect product.

It is useful, but not directly relevant, that the centralizer of λ(G) in Sym(G) is ρ(G), their intersection is ρ(Z(G)) = λ(Z(G)), where Z(G) is the center of G, and that A is a common complement to both of these normal subgroups of N.

Properties

ρ(G) ∩ Aut(G) = 1

Aut(G) normalizes ρ(G) so that canonically ρ(G)Aut(G) ≅ G ⋊ Aut(G)

Inn ⁡ ( G ) ≅ Im ⁡ ( g ↦ λ ( g ) ρ ( g ) ) since λ(g)ρ(g)(h) = ghg⁻¹

K ≤ G is a characteristic subgroup if and only if λ(K) ⊴ Hol(G)