On the pronormality of subgroups of odd index in some direct products of finite groups

Abstract

A subgroup [Formula: see text] of a group [Formula: see text] is said to be pronormal in [Formula: see text] if [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text] for each [Formula: see text]. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product [Formula: see text] of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of [Formula: see text]. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text]. Thus, in this paper, we obtain a criterion of pronormality of a subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] is a prime and each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text].