This question already has an answer here:

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • 2
    $\begingroup$ If by sum you mean direct product then iff $G$ is nilpotent $\endgroup$ – user8268 May 18 '17 at 16:11
  • $\begingroup$ A remark: Direct sum is only defined for abelian groups. For finite abelian groups, direct sum is the same as direct product. See Direct sum of finitely-many non-abelian groups for generalization to nonabelian case. $\endgroup$ – Alex Vong May 18 '17 at 16:13
  • $\begingroup$ Yes I realize that sum is used for abelian cases. I really do mean product. $\endgroup$ – Scotty Vol May 18 '17 at 16:14
  • $\begingroup$ Thank you. Nilpotent is what I am looking for. $\endgroup$ – Scotty Vol May 18 '17 at 16:37
  • $\begingroup$ To answer your last question, a direct product of subgroups is abelian if and only if each of those subgroups is abelian. So a finite nilpotent group is abelian if and only if all of its Sylow subgroups are abelian. $\endgroup$ – Bungo May 18 '17 at 17:43