# Internal semidirect product

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This article describes a product notion for groups. See other related product notions for groups.

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## Definition

### Definition with symbols

A group is termed an **internal semidirect product** of subgroups and if the following hold:

- is a normal subgroup of
- and are permutable complements

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

### Equivalence with external semidirect product

`Further information: Equivalence of internal and external semidirect product`

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

## 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.

## Examples

### 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 and , is the internal semidirect product of and . 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 and .
- The symmetric group of degree four is an internal semidirect product of the normal subgroup and a six-element subgroup (isomorphic to symmetric group of degree three) comprising the permutations that fix .
- The dihedral group of degree and order is the internal semidirect product of a cyclic subgroup of order (the
*rotations*) and a cyclic subgroup of order (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.

## Relation with other properties

### Stronger product notions

### Weaker product notions

### 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 of a group with two complements and , it is
*not*necessary that there exist an isomorphism of sending to . It is also not necessary that the actions of and on 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.