Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that:

Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\phi\colon H \to \text{Aut}(G)$, then $$G\rtimes_\phi H \;=\; \langle X, Y \mid R,\,S,\,yxy^{-1}=\phi(y)(x)\text{ for all }x\in X\text{ and }y\in Y\rangle.$$

Specifically I have seen it on stack exchange, and on group props, however neither of these sources have a proof, or any references.

I have searched all over for a references to a book or paper, however I have yet to find any.

To be more clear: I am looking for proof that the group formed by the semidirect product $G\rtimes_\phi H$ is isomorphic to the group given by the presentation $$\langle X, Y \mid R,\,S,\,yxy^{-1}=\phi(y)(x)\text{ for all }x\in X\text{ and }y\in Y\rangle.$$

Does anyone have any ideas? Thanks!

• Hmm, not sure of a reference, as it is just immediate from the definition. Feb 5, 2016 at 22:03
• What do you mean by a proof? A proof that it is a group? That is a straightforward exercise. Feb 5, 2016 at 22:03
• I understand that the presentation defines a group upto isomorphism, however how can one show that group is isomorphic to the semi direct product? Feb 5, 2016 at 22:10
• It is proved in Section 10.2 of the book "Presentations of Groups" by D.L. Johnson, The result proved there is more general and covers all extensions, not just semidirect products. But the proof is not exactly difficult! Feb 5, 2016 at 22:42
• @DerekHolt: I can find neither Section 10.2 nor that presentation in Section 10 of that book. Jan 12, 2018 at 17:22