Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 9E / b / Solution

Let $S$ act by conjugation on the Sylow $p$-subgroups. The only fixed point is $S$: if $S$ normalizes another $T$, then $ST$ is a $p$-subgroup, forcing $S=T$. For $T=gS{g}^{-1}\ne S$, a stabilizer element in $S$ normalizes $T$, hence lies in $T$ by the same argument; the hypothesis then makes it trivial. Every nontrivial orbit therefore has size $|S|$, so