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If every Sylow subgroup of G is a normal subgroup, show that G is isomorphic to the product of its Sylow subgroups.

I can't figure this out without G being abelian.

Nicky Hekster
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Mertle
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  • Hint: Let $H_1, H_2, \ldots, H_k$ be the distinct normal Sylow subgroups, then check that $$ H_i\cap \prod_{j\neq i} H_j = {e} $$ – Prahlad Vaidyanathan Dec 11 '13 at 07:50
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    First question you should answer is : $\textbf{when would $G\cong H\times K$ for subgroups $H,K$ of $G$? }$ –  Dec 11 '13 at 12:04
  • Thank you. I have done both of these things. G is isomorphic to the product of H and K when G is abelian, the intersection of the subgroups is only the identity, and HK=G. – Mertle Dec 11 '13 at 19:02
  • @Mertle : you have to prove $G$ is abelian then do you think it makes any sense to say "G is isomorphic to the product of H and K when G is abelian," –  Dec 12 '13 at 05:54

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