Internal semidirect product

Definition with symbols
A group $$G$$ is termed an internal semidirect product of subgroups $$N$$ and $$H$$ if the following hold:


 * $$N$$ is a normal subgroup of $$G$$
 * $$N$$ and $$H$$ are permutable complements

Note here that $$H$$ acts as automorphisms on $$N$$ by the conjugation action.

Equivalence with external semidirect product
Suppose $$G$$ is an internal semidirect product with normal subgroup $$N$$ and $$H$$ as the other subgroup. If we start out with $$N$$ and $$H$$ as abstract groups, and with the action of $$H$$ on $$N$$ (abstractly) which comes from the conjugation in $$G$$, then the external semidirect product formed from these is isomorphic to $$G$$.

Terminology

 * A subgroup which occurs as the normal subgroup for an internal semidirect product is termed a complemented normal subgroup, sometimes also called split normal subgroup.
 * A subgroup which occurs as the permutable complement to a normal subgroup, is termed a retract. This is because there is a retraction from the whole group, to this subgroup, whose kernel is the normal subgroup.

Trivial examples

 * Every group is the internal semidirect product of itself and the trivial subgroup. In fact, it is an internal direct product of itself and the trivial subgroup.
 * Given two groups $$G$$ and $$H$$, $$G \times H$$ is the internal semidirect product of $$G \times \{ e \}$$ and $$\{ e \} \times H$$. In fact, it is the internal direct product.

Simple examples

 * The symmetric group on any finite set of size at least two is the internal semidirect product of the alternating group and the two-element subgroup generated by any transposition. For instance, the symmetric group of degree three is the internal semidirect product of the subgroups $$\{, (1,2,3), (1,3,2)\}$$ and $$\{ , (1,2) \}$$.
 * The symmetric group of degree four is an internal semidirect product of the normal subgroup $$\{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$ and a six-element subgroup (isomorphic to symmetric group of degree three) comprising the permutations that fix $$4$$.
 * The dihedral group of degree $$n$$ and order $$2n$$ is the internal semidirect product of a cyclic subgroup of order $$n$$ (the rotations) and a cyclic subgroup of order $$2$$ (generated by a reflection).

Non-examples

 * The cyclic group of order four has a normal subgroup of order two, but this normal subgroup has no complement.

Stronger product notions

 * Internal regular product
 * Internal direct product

Weaker product notions

 * Exact factorization
 * Group extension

Related subgroup properties

 * Complemented normal subgroup is a normal subgroup having a permutable complement, and hence, part of a semidirect product.
 * Retract is a subgroup having a normal complement, and hence, part of a semidirect product.

Related group properties

 * Splitting-simple group is a group that cannot be expressed as an internal semidirect product of nontrivial subgroups.

Facts

 * Complement to normal subgroup is isomorphic to quotient: In particular, given the normal subgroup for a semidirect product, we know the isomorphism type of the complement.
 * Complements to normal subgroup need not be automorphic: Given a normal subgroup $$N$$ of a group $$G$$ with two complements $$H_1$$ and $$H_2$$, it is not necessary that there exist an isomorphism of $$G$$ sending $$H_1$$ to $$H_2$$. It is also not necessary that the actions of $$H_1$$ and $$H_2$$ on $$N$$ are the same.
 * Complements to abelian normal subgroup are automorphic
 * Complements to normal Hall subgroup are conjugate: This is the conjugacy part of the result commonly known as the Schur-Zassenhaus theorem.