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 $.