Reference for decomposition in invariants and derived subgroup in a semidirect product of abelian groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:14:49Z http://mathoverflow.net/feeds/question/70416 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70416/reference-for-decomposition-in-invariants-and-derived-subgroup-in-a-semidirect-pr Reference for decomposition in invariants and derived subgroup in a semidirect product of abelian groups Maurizio Monge 2011-07-15T09:24:08Z 2011-07-15T16:21:34Z <p>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 $$A = C \oplus G'.$$</p> <p>Is there a quick reference for this fact? Please note that i'm <b>NOT</b> 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!</p> http://mathoverflow.net/questions/70416/reference-for-decomposition-in-invariants-and-derived-subgroup-in-a-semidirect-pr/70448#70448 Answer by Derek Holt for Reference for decomposition in invariants and derived subgroup in a semidirect product of abelian groups Derek Holt 2011-07-15T16:21:34Z 2011-07-15T16:21:34Z <p>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.</p>