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

From Groupprops

## Contents

## Statement

### In terms of external semidirect product

Suppose is a group and is a group acting on , i.e., a homomorphism of groups is specified. Suppose we are given presentations for both and , and assume further that the generating sets for and have already been *made disjoint*, i.e., no generator letter is repeated between the two presentations.

The External semidirect product (?) can be given the following presentation:

- Generating set is taken as the union of generating sets for and .
- Relation set is taken as the union of relation sets for and and the following
*action*relations: for every generator of and every generator of , the relation where is the word (in ) for the element ).

### In terms of internal semidirect product

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