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

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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Related facts

• Presentation of free product is disjoint union of presentations
• Presentation of direct product is disjoint union of presentations plus commutation relations