TY - JOUR

T1 - On the pronormality of subgroups of odd index in some direct products of finite groups

AU - Maslova, N. V.

AU - Revin, D. O.

N1 - This work was supported by the Russian Science Foundation (project 19-71-10067). Publisher Copyright: © 2023 World Scientific Publishing Company.

PY - 2023/4/1

Y1 - 2023/4/1

N2 - A 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 i=1t 3Symni, where each Symni acts naturally on {1,hellip,ni}, such that H projects onto i=1tSym ni. Thus, in this paper, we obtain a criterion of pronormality of a subgroup H of odd index in a subgroup of i=1 piSymni, where each pi is a prime and each Symni acts naturally on {1,...,ni}, such that H projects onto i=1tSym ni.

AB - A 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 i=1t 3Symni, where each Symni acts naturally on {1,hellip,ni}, such that H projects onto i=1tSym ni. Thus, in this paper, we obtain a criterion of pronormality of a subgroup H of odd index in a subgroup of i=1 piSymni, where each pi is a prime and each Symni acts naturally on {1,...,ni}, such that H projects onto i=1tSym ni.

KW - direct product

KW - Finite group

KW - odd index

KW - pronormal subgroup

KW - simple symplectic group

KW - wreath product

UR - http://www.scopus.com/inward/record.url?scp=85124829305&partnerID=8YFLogxK

UR - https://elibrary.ru/item.asp?id=48150826

UR - https://www.elibrary.ru/item.asp?id=48150826

UR - https://www.mendeley.com/catalogue/87519bef-c755-35f3-b5dc-f616a5d5a346/

U2 - 10.1142/S0219498823500834

DO - 10.1142/S0219498823500834

M3 - Article

AN - SCOPUS:85124829305

VL - 22

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

SN - 0219-4988

IS - 4

M1 - 2350083

ER -