# When is $G$, a finite group, the sum of its Sylow $p$-subgroups? [duplicate]

Questions like this have been asked before, but I just want some quick clarification on something.

Suppose that $G$ is the sum of its Sylow $p$-subgroups. Then each Sylow subgroup must be normal, and of course we have the other direction. If each Sylow subgroup is normal, then $G$ is the product of its Sylow subgroups.

My question that I need clarified is, "if $G$ is a sum of its Sylow subgroups, when will this be abelian, non-abelian?"

## marked as duplicate by Dietrich Burde, Community♦May 18 '17 at 18:31

• If by sum you mean direct product then iff $G$ is nilpotent – user8268 May 18 '17 at 16:11