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Let $G$ be a finite group with commutator subgroup $G'$. Let $p$ be a prime number. Then $p \nmid |G'|$ if and only if $G$ has an abelian Sylow $p$-subgroup $P$ and normal $p$-complement $N$ (and in this case $G=N \rtimes P$ ).

This is surely a standard result, but I can't seem to find a proof written down anywhere. I have worked out a straightforward proof below, but if someone could provide a good reference to a proof (so that I can just cite it in a paper I'm writing), that would be great.

Proof: Suppose that $G$ has an abelian Sylow $p$-subgroup $P$ with normal complement $N$. Then $P \simeq G/N$ is abelian so $G' \leq N$. But $p \nmid |N|$ so $p \nmid |G'|$. Suppose conversely that $p \nmid |G'|$. Let $P$ be a Sylow $p$-subgroup of $G$ and let $\theta:G \rightarrow G/G'$ be the natural projection. Then $P \cap G'$ is trivial and so $\theta$ restricted to $P$ is an isomorphism. But $G/G'$ is abelian, so $P$ is abelian. Since $G/G'$ is abelian, there exists a subgroup $M$ such that $G/G' = \theta(P) \times M$. Let $\psi:G/G' \rightarrow \theta(P)$ be the natural projection. Then put $N:=\ker(\psi \circ \theta)$.

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This is obvious, so you do not really need a reference. You can just say (if you are writing a paper) that the result is most probably well known and you give a proof only for the sake of completeness. –  Mark Sapir Mar 9 '12 at 14:49
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Yes, it's an elementary application of isomorphism theorems- the second part of your proof could be shortened. –  Geoff Robinson Mar 9 '12 at 15:52
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The second part may be shorted as follows: Let $P$ be a sylow $p$-subgroup of $G$. Since $(p, |G'|)=1$, $PG'/G'$ is a Sylow $p$-subgroup of the abelian group $G/G'$, and then $G/G'=PG'/G' \times N/G'$. Thus $G=PN$ and $P \cap N=1$.

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@Henri: The hypothesis that the prime $p$ divides $|G|$ is crucial in the statement of the result. It should be started something like as follows: Let $G$ be a finite group and $p$ be a prime dividing the order of $G$. –  Alireza Abdollahi Mar 30 '12 at 7:28
    
@Alireza: good point - I was being a bit sloppy! –  Henri Johnston May 4 '12 at 15:14
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