So Sylow's second theorem tells us that if G is a group, p a prime, and H,K are both p-Sylow subgroups, then [itex] \exists g \in G [/itex] such that [itex] H = gKg^{-1} [/itex]. Now I have some questions about this, because I don't think I've ever properly understood the consequences of this theorem.(adsbygoogle = window.adsbygoogle || []).push({});

My issue I think is really the act of conjugation. Namely, this theorem states that we can always relate two p-Sylow subgroups via conjugation, but is it really more that conjugation is a transitive group action on the set of p-Sylow subgroups?

This may sound silly, but we always say that if there is precisely one p-Sylow subgroup, it must be normal right? But the way that I've always read this is that we can only be assured that there is a single element [itex] g \in G [/itex] such that [itex] gPg^{-1} = P [/itex] and that this needn't hold for all [itex] g \in G[/itex]. So I guess what I'm asking is, does conjugation of a p-Sylow subgroup ALWAYS yield another p-Sylow subgroup? This would then clear up a lot of my confusion.

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# Easy question on the Second Sylow Theorem

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