Holomorph (mathematics)

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group $$G$$, 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 $$\phi:A\rightarrow \operatorname{Aut}(G)$$ 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:
 * $$(x^{i_1},\sigma^{j_1})(x^{i_2},\sigma^{j_2}) = (x^{i_1+i_22^{^{j_1}}},\sigma^{j_1+j_2})$$ where the exponents of $$x$$ are taken mod 3 and those of $$\sigma$$ mod 2.

Observe, for example
 * $$(x,\sigma)(x^2,\sigma)=(x^{1+2\cdot2},\sigma^2)=(x^2,1)$$

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), $$\lambda_g$$(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).
 * $$\lambda_x$$(1) = x·1 = x,
 * $$\lambda_x$$(x) = x·x = x2, and
 * $$\lambda_x$$(x2) = 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·$$\lambda_g$$ = $$\lambda_{n(g)}$$·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·$$\lambda_g$$·$$\lambda_h$$ and once to the (equivalent) expression n·$$\lambda_{gh}$$ 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 $$\lambda_G$$, 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 $$\rho_{Z(G)}=\lambda_{Z(G)}$$, 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)
 * $$\operatorname{Inn}(G)\cong \operatorname{Im}(g\mapsto \lambda(g)\rho(g))$$ since λ(g)ρ(g)(h) = ghg&minus;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)