An example of a relatively maximal non-pronormal subgroup of odd order in a finite simple group (Q6610188)

A subgroup \(H\) of a group \(G\) is pronormal if \(H\) and \(H^{g}\) are conjugate in \(\langle H, H^{g} \rangle\) for all \(g \in G\). It is clear that every maximal and every Sylow subgroup is pronormal. A class \(\mathfrak{X}\) of groups closed under subgroups, homomorphic images, and extensions is called a closed class and an \(\mathfrak{X}\)-maximal subgroup \(H\) of \(G\) is a subgroup \(H \leq G\) maximal with respect to \(H \in \mathfrak{X}\). Typical examples of complete classes are the class of soluble groups and the class of \(\pi\)-groups (\(\pi\) a set of prime numbers).\N\NIn this paper, the author investigates the problem of deciding whether every subgroup \(H\) that is \(\mathfrak{X}\)-maximal in a finite simple group \(G\) is necessarily pronormal. The main result is that the group \(G= \mathrm{Sp}_{4}(4)\) has a non-pronormal \(\{3, 5\}\)-maximal subgroup of order 15.