In mathematics, especially in the area of algebra known as group theory, the holomorph of a group G, denoted, is a group that simultaneously contains (copies of) G and its automorphism group. 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 is the automorphism group of G then where the multiplication is given by Typically, a semidirect product is given in the form where G and A are groups and is a homomorphism and where the multiplication of elements in the semidirect product is given as which is well defined, since and therefore. For the holomorph, and \phi is the identity map, as such we suppress writing \phi explicitly in the multiplication given in equation 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 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), \lambda_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 = {1, x, x2 } is a cyclic group of order three, then 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·\lambda_g = \lambda_h·n. If an element n of the holomorph fixes the identity of G, then for 1 in G, (n·\lambda_g)(1) = (\lambda_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·\lambda_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·\lambda_g·\lambda_h and once to the (equivalent) expression n· 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 \lambda_G, and the only \lambda_g that fixes the identity is λ(1). Setting A to be the stabilizer of the identity, the subgroup generated by A and \lambda_G is semidirect product with normal subgroup \lambda_G and complement A. Since \lambda_G is transitive, the subgroup generated by \lambda_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 \lambda_G in Sym(G) is \rho_G, their intersection is, where Z(G) is the center of G, and that A is a common complement to both of these normal subgroups of N.

Properties