An Example of a Relatively Maximal Nonpronormal Subgroup of Odd Order in a Finite Simple Group

Abstract

We prove the existence of a triple \( ({\mathfrak{X}},G,H) \), where \( {\mathfrak{X}} \) is a class of finite groups consisting of groups of odd order which is complete (i.e., closed under subgroups, homomorphic images, and extensions), \( G \) is a finite simple group, \( H \) is an \( {\mathfrak{X}} \)-maximal subgroup in \( G \), and \( H \) is not pronormal in \( G \).