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I would like to know if there is some results concerning about the following question:

When could a $p$-Sylow subgroup of a finite ring $R$ be a subring?

In other words, is it possible to induce the multiplication of the ring on the $p$-Sylow? If yes, are there conditions to guarantee this?

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    No, I started to think about this just today. I am looking for. Let's continue.2012-08-21

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For $r \in R$ fixed, the maps $s \mapsto rs$ and $s \mapsto sr$ are endomorphisms of the abelian group structure on $R$. Now use the fact that for an abelian group $G$ with Sylow subgroup $P$, any endomorphism $\phi$ of $G$ stabilizes $P$: $\phi(P) \subseteq P$ (this follows from standard Sylow theory). Therefore the Sylow subgroups are actually (two sided) ideals in $R$.

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    @Sigur Yes, sure.2012-08-21