Let $A$ and $B$ be finite abelian groups with coprime order, and let $G=A\rtimes{}B$ be a semidirect product, via any action. Let $C\subseteq{}A$ be the subgroup of the elements of $A$ which are fixed by the action of $B$, so that $C=Z(G)\cap{}A$. Then we have
$$
 A = C \oplus G'.
$$


Is there a quick reference for this fact? Please note that i'm **NOT** asking for a proof of this simple (and well known i guess?) fact, i just need a reference to quickly point to in a note, to avoid making it cumbersome. Unless there is a one-line proof that i missed.
Thanks for the attention!



Reference for decomposition in invariants and derived subgroup in a semidirect product of abelian groups
Theorem 2.3 in Chapter 5 of the book "Finite Groups" by Daniel Gorenstein states that if $A$ is a $p'$-group of automorphisms of an abelian $p$-group $P$, then $P = C\_P(A) \times [P,A]$ (all groups here are assumed to be finite). You can deduce your result easily from this.


