In mathematics, especially in the area of algebra known as group theory, the **holomorph** of a group is a group that 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 semidirect product

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 semidirect 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 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(*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* = *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*(*g*·*h*) 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 *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

*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}( is the group of inner automorphisms of*G*.) *K*≤*G*is a characteristic subgroup if and only if*λ*(*K*) ⊴ Hol(*G*)

## References

- Hall, Marshall, Jr. (1959),
*The theory of groups*, Macmillan, MR 0103215 - Burnside, William (2004),
*Theory of Groups of Finite Order, 2nd ed.*, Dover, p. 87