- #1

Kreizhn

- 743

- 1

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.