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