Presentation of semidirect product is disjoint union of presentations plus action by conjugation relations

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Statement

In terms of external semidirect product

Suppose N is a group and H is a group acting on N, i.e., a homomorphism of groups \rho:H \to \operatorname{Aut}(N) is specified. Suppose we are given presentations for both N and H, and assume further that the generating sets for N and H have already been made disjoint, i.e., no generator letter is repeated between the two presentations.

The External semidirect product (?) N \rtimes H can be given the following presentation:

  • Generating set is taken as the union of generating sets for N and H.
  • Relation set is taken as the union of relation sets for N and H and the following action relations: for every generator a of N and every generator b of N, the relation bab^{-1} = w where w is the word (in N) for the element \rho(b) \cdot a).

In terms of internal semidirect product

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