Finite groups with few conjugacy classes of $p$-subgroups

Abstract

Let $G$ be a finite group, and let $p$ be an odd prime of $|G|$ and Sylow $p$-subgroups of $G$ be non-abelian. In this paper, we prove that any two $p$-subgroups of equal order in $G$ are conjugate if and only if $G/O_{p^{\prime }}(G)$ is isomorphic to ${}^2{F}_4(2)$ and $p=3$, $\mathrm{Ru}$ and $p=3$, $J_4$ and $p=3$, $\mathrm{Th}$ and $p=5$, or an almost simple group of the socle ${}^2{F}_4(2^{2n+1})$ with $n>0,n\not\equiv 1\mathrm{mod}3$ and $p=3$. This solved a problem of Brandl’s.