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

A subgroup H of a group G is said to be pronormal in G if H and H-g are conjugate in < H, H-g > for each g is an element of 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 Pi(t)(i=1) Z(3) (sic) Sym(ni), where each Sym(ni) acts naturally on {1, ..., n(i)}, such that H projects onto Pi(t)(i=1) Sym(ni). Thus, in this paper, we obtain a criterion of pronormality of a subgroup H of odd index in a subgroup of Pi(t)(i=1) Z(3) (sic) Sym(ni), where each pi is a prime and each Sym(ni) acts naturally on {1, ..., n(i)}, such that H projects onto Pi(t)(i=1) Sym(ni).