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External semidirect product

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Definition

Suppose N is a group and H is a group acting on N; in other words, there is a group homomorphism 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') = (ab.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
  • 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.

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

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