I have the following question: In the context of a finite supersolvable group $G$, where $p$ represents the largest prime divisor of $|G|$, I would like to inquire about a specific theorem, for which I am seeking a reference. The theorem concerns the behavior of $p'$-elements in such groups. Specifically, if the $p'$-elements of $G$ act as power automorphisms on the Sylow $p$-subgroup of $G$, does this theorem, attributed to Huppert (reference needed), state that either the Sylow $p$-subgroup $P$ must be abelian or that the $p'$-elements of $G$ centralize $P$?
If anyone can provide a reference for this theorem, I'd greatly appreciate it.
Thank you for your assistance!
Maybe this helps: on page $34$ in Roland Schmidt's book, Subgroup Lattices of Groups, de Gruyter 1994, (and here) Exercise, no.5. It reads: If $P$ is a non-abelian $p$-group, then the group of power automorphisms of $P$ is a $p$-group. Reference is made to an article by Huppert, in Archiv Math., 12, (1961), pp. 161-169, entitled: Zur Sylowstruktur auflösbarer Gruppen. In the case you mention, it concerns a supersolvable group $G$ where $p$ is the largest prime dividing $|G|$. Such a $G$ has a normal Sylow $p$-subgroup. Incidentally, the group of power automorphisms of a finite group is abelian. That's in Schmidt's book Theorem I.5.1. Hope this helps!