I believe there is a straightforward formula for the abelianization of a semi-direct product: if $G$ acts on $H$, and we form the semi-direct product of $G$ and $H$ in the usual way, and the abelianization of this semi-direct product is the product $G^{ab}\times (H^{ab})\_{G}$.


(Here the subscript $G$ denotes taking the coinvariants with respect to $G$. That is, $(H^{ab})\_{G}$ is a the quotient of $H^{ab}$ by the subgroup generated by elements of the form $h^g-h$ for $h$ in $H$ and $g$ in $G$, and where the superscript $g$ denotes the action of $G$ on $H^{ab}$ induced by the action of $G$ on $H$.)


Does anyone happen to know a good reference for this?



Abelianization of a semidirect product
I agree with Ryan and Victor, except that you don't need presentations. The subgroup $[G \ltimes H,G \ltimes H]$ is generated by $[H,H] \cup [G,H] \cup [G,G]$, so you can write
$$(G \ltimes H)^{ab} = (G \ltimes H) / \langle [H,H] \cup [G,H] \cup [G,G] \rangle.$$
If you apply the relators $[H,H]$, you get $G \ltimes H^{ab}$; then if you apply the relators $[G,H]$, you get $G \times (H^{ab})\_G$; then finally if you apply $[G,G]$, you get $G^{ab} \times (H^{ab})\_G$. You can add this as an extra half-paragraph or footnote rather than giving a citation.


I don't think that the referee has the right to demand a longer explanation than this, unless maybe you are writing a textbook.


