Prime-set decomposition of elementary groups ($E=E_{\pi }\times E_{{\pi }^{\prime }}$)

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.