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

Abstract

subgroup H of a group G is said to be pronormal in G if H and Hg are conjugate in (H,Hg) for each g ∈ G. 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 G 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 G. 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 H of odd index in a subgroup of Qt i=1 Z3 Symni , where each Symni acts naturally on {1, . . . ,ni}, such that H projects onto Qt i=1 Symni . Thus, in this paper, we obtain a criterion of pronormality of a subgroup H of odd index in a subgroup of Qt i=1 Zpi Symni , where each pi is a prime and each Symni acts naturally on {1, . . . , ni}, such that H projects onto Qt i=1 Symni . © 2023 World Scientific Publishing Company.