Prime-set decomposition of elementary groups
= Prime-set decomposition of elementary groups
{title2=$E=E_\pi\times E_{\pi'}$}
Write an <elementary group> as $P\times C$ with $P$ a p-group and $C$ cyclic of order prime to $p$. Split $C$ into its cyclic pi-part and pi-complementary part. Attach $P$ to whichever part contains $p$, producing a <direct product of groups> of a <pi-group> and a complementary-prime group.