Holomorph (mathematics)


In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group, the holomorph of denoted can be described as a semidirect product or as a permutation group.

Hol(''G'') as a semi-direct product

If is the automorphism group of then
where the multiplication is given by
Typically, a semidirect product is given in the form where and are groups and is a homomorphism and where the multiplication of elements in the semi-direct product is given as
which is well defined, since and therefore.
For the holomorph, and is the identity map, as such we suppress writing explicitly in the multiplication given in above.
For example,
Observe, for example
and this group is not abelian, as, so that is a non-abelian group of order 6, which, by basic group theory, must be isomorphic to the symmetric group.

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, λ = 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 is defined by ρ = h·g−1, where the inverse ensures that ρ = ρ). 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
so λ takes to.
The image of λ is a subgroup of Sym isomorphic to G, and its normalizer in Sym 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 = λ·n. If an element n of the holomorph fixes the identity of G, then for 1 in G, =, but the left hand side is n, 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 = λ·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·λ and once to the expression n·λ gives that n·n = n. That is, every element of N that fixes the identity of G is in fact an automorphism of G. Such an n normalizes λ, and the only λ that fixes the identity is λ. Setting A to be the stabilizer of the identity, the subgroup generated by A and λ is semidirect product with normal subgroup λ and complement A. Since λ is transitive, the subgroup generated by λ 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 in Sym is ρ, their intersection is ρ = λ, where Z is the center of G, and that A is a common complement to both of these normal subgroups of N.

Properties