Holomorph (mathematics) explained

G , denoted \operatorname{Hol}(G) , is a group that simultaneously contains (copies of) G and its automorphism group \operatorname{Aut}(G) . It provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. The holomorph can be described as a semidirect product or as a permutation group.

Hol(G) as a semidirect product

If

\operatorname{Aut}(G) is the automorphism group of G then\operatorname{Hol}(G) = G\rtimes \operatorname{Aut}(G) where the multiplication is given by

Typically, a semidirect product is given in the form

G\rtimes _{\phi} A where G and A are groups and \phi:A → \operatorname{Aut}(G) is a homomorphism and where the multiplication of elements in the semidirect product is given as(g,a)(h,b) = (g\phi(a)(h),ab) which is well defined, since \phi(a)\in \operatorname{Aut}(G) and therefore \phi(a)(h)\in G .

For the holomorph,

A = \operatorname{Aut}(G) and \phi is the identity map, as such we suppress writing \phi explicitly in the multiplication given in equation above.

For example,

G = C _{3=\langle} x\rangle = \{ 1 ,x,x ^{2\}} the cyclic group of order 3\operatorname{Aut}(G) = \langle \sigma\rangle = \{ 1 ,\sigma\} where \sigma(x) = x ^{2} \operatorname{Hol}(G) = \{(x ^{i,\sigma} ^{j)\}} with the multiplication given by:(x ^{i _{1}} ,\sigma ^{j _{1}} )(x ^{i _{2}} ,\sigma ^{j _{2}} ) = (x ^{i _{1+ i} ^{^{j _{1}}} ^{22}} ,\sigma ^{j _{1+j} _{2}} ) where the exponents of x are taken mod 3 and those of \sigma mod 2.

Observe, for example

(x,\sigma)(x ^{2,\sigma)=(x} ^{1+2 ⋅ 2} ,\sigma ^{2)=(x} ^{2,1)} and this group is not abelian, as (x ^{2,\sigma)(x,\sigma)=(x,1)} , so that \operatorname{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 \rho _{g} (h) = h·g⁻¹, where the inverse ensures that \rho _{gh} (k) = \rho _{g} (\rho _{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 = is a cyclic group of order three, then

λ _{x} (1) = x·1 = x,λ _{x} (x) = x·x = x², andλ _{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 _{gh} 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 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 \rho _{G} , their intersection is \rho _{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)

\operatorname{Inn}(G)\cong \operatorname{Im}(g\mapsto λ (g)\rho(g)) since λ(g)ρ(g)(h) = ghg^-1 (\operatorname{Inn}(G) is the group of inner automorphisms of G.)

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