External semidirect product

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Definition

Definition with the left action convention

Suppose N is a group and H is a group acting on N; in other words, there is a group homomorphism \rho:H \to \operatorname{Aut}(N), from H to the automorphism group of N. The external semidirect product G of N and H, denoted N \rtimes H is, as a set, the Cartesian product N \times H, with multiplication given by the rule:

\! (a,b)(a',b') = (a(\rho(b)(a')),bb')

Writing the action \rho(b)a' = b \cdot a', we get:

(a,b)(a',b') = (a(b \cdot a'),bb')

The way multiplication is defined, it turns out that:

  • N embeds as a normal subgroup of G (via a \mapsto (a,e)) and H embeds as a subgroup via b \mapsto (e,b). The two subgroups are permutable complements, hence the external semidirect product is the same as an internal semidirect product once we identify N and H with their images in G. In particular, the image of N is a complemented normal subgroup in G and the image of H is a retract of G.
  • The action of the image of H, on the image of N, via conjugation in G, is the same as the abstract action that we started with.

Comments

Case of abelian normal subgroup

In the special case where N is an abelian group and the binary operation of N is denoted additively, the multiplication rule for G can be written as:

\! (a,b)(a',b') = (a + (b \cdot a'), bb')

This notation comes up in the study of the second cohomology group.

Case of trivial action

The external semidirect product becomes an external direct product when the action of H on N is trivial.

Related notions

Related notions for groups

Generalizations to other algebraic structures