Internal semidirect product

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


This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

Definition with symbols

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

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

Equivalence with external semidirect product

Further information: Equivalence of internal and 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.

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 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

Non-examples

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