On pronormal subgroups in finite simple groups (Q1709840)

A subgroup $H$ of a group $G$ is called pronormal if for every $g\in G$, the groups $H$ and $H^g$ are conjugate in the subgroup they generate. In particular, normal subgroups and maximal subgroups are pronormal, as are Hall subgroups of solvable groups. The conjecture that subgroups of odd index in simple groups are pronormal turns out to be false, and certain symplectic groups are counterexamples. However, for `most' simple groups, the conjecture is true, so the question then is to determine exactly which simple groups have this property. In this paper, the authors sketch a proof that subgroups of odd order are pronormal in $\mathrm{PSp}_{2n}(q)$ for $q\equiv \pm 3 \mod 8$ and $n$ of the form $2^m(2^{2k}+1)$, where $m\ge 0$ and $k\ge 1$.