On the pronormality of subgroups of odd index in finite simple groups. (Q259844)

By definition a subgroup \(H\) of a group \(G\) is called pronormal in \(G\) if for every element \(g\) of \(G\), the subgroups \(H\) and \(H^g\) are conjugate in \(\langle H,H^g\rangle\). It is conjectured [\textit{E. P. Vdovin} and \textit{D. O. Revin}, Sib. Math. J. 53, No. 3, 419-430 (2012; Zbl 1275.20008); translation from Sib. Mat. Zh. 53, No. 3, 527-542 (2012)] that the subgroups of odd index of a finite simple group are pronormal. The aim of the paper under review is to prove the above conjecture for a family of simple groups such as the alternating groups, groups of Lie type over a field of characteristic 2, etc.