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

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 fact about::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 $$\rho(b) \cdot a$$).

Related facts

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