The Sylow theorems say that all $p$-Sylow subgroups are conjugate.
Is that the only thing they are conjugate to? In other words could a $p$-Sylow subgroup also be conjugate to something that is not a $p$-Sylow group?
Or does the fact that every element with order dividing $p$ is in a $p$-Sylow group mean that that is never possible?
Best Answer
To get this question an answer, here is Jyrki Lahtonen's answer:
So p-Sylow subgroups are always isomorphic to other p-Sylow subgroups.