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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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    It is certainly (and trivially) true for finite fields, so yes, it is possible (and being a field is an example of a condition that would guarantee this).2012-08-21
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    @tomasz. So the interesting cases are for non fields. Any idea?2012-08-21
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    Actually, do you have any examples when it is *not* true?2012-08-21
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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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