# 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 (1) 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:
    $(x^{i_{1}},\sigma ^{j_{1}})(x^{i_{2}},\sigma ^{j_{2}})=(x^{i_{1}+i_{2}2^{^{j_{1}}}},\sigma ^{j_{1}+j_{2}})$ where the exponents of $x$ are taken mod 3 and those of $\sigma$ mod 2.
Observe, for example
    $(x,\sigma )(x^{2},\sigma )=(x^{1+2\cdot 2},\sigma ^{2})=(x^{2},1)$
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 ρ$_g$(h) = h·g$^{−1}$, where the inverse ensures that ρ$_{gh}$(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 = C$_3$ = {1, x, x$^2$ } is a cyclic group of order three, then
* λ$_x$(1) = x·1 = x,
* λ$_x$(x) = x·x = x$^2$, and
* λ$_x$(x$^2$) = x·x$^2$ = 1,
so λ(x) takes (1, x, x$^2$) to (x, x$^2$, 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$_{gg}$ 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 ρ$_G$, 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$^{−1}$ ($\operatorname {Inn} (G)$ is the group of inner automorphisms of G.)
* K ≤ G is a characteristic subgroup if and only if λ(K) ⊴ Hol(G)