Holomorph (mathematics)

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In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group which simultaneously contains (copies of) 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[edit]

If is the automorphism group of then

where the multiplication is given by

[Eq. 1]

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 [Eq. 1] above.

For example,

  • the cyclic group of order 3
  • where
  • with the multiplication given by:
where the exponents of are taken mod 3 and those of mod 2.

Observe, for example

and note also that 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[edit]

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(hn. If an element n of the holomorph fixes the identity of G, then for 1 in G, (n·λ(g))(1) = (λ(hn)(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(gλ(h) and once to the (equivalent) expression n·λ(g·h) gives that n(gn(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[edit]

  • ρ(G) ∩ Aut(G) = 1
  • Aut(G) normalizes ρ(G) so that canonically ρ(G)Aut(G) ≅ G ⋊ Aut(G)
  • since λ(g)ρ(g)(h) = ghg−1
  • KG is a characteristic subgroup if and only if λ(K) ⊴ Hol(G)

References[edit]