TY - JOUR

T1 - On the pronormality of subgroups of odd index in finite simple symplectic groups

AU - Kondrat’ev, A. S.

AU - Maslova, N. V.

AU - Revin, D. O.

N1 - Кондратьев А.С., Маслова Н.В., Ревин Д.О. О пронормальности подгрупп нечетных индексов в конечных простых симплектических группах // Сибирский математический журнал. – 2017. – Т. 58. – № 3(343). – С. 599-610. Работа выполнена при финансовой поддержке Совета по грантам Президента РФ (проект МК–6118.2016.1) и Программы государственной поддержки ведущих университетов РФ (соглашение № 02.A03.21.0006 от 27.08.2013). Второй автор является стипендиатом Фонда Дмитрия Зимина «Династия» (программа поддержки молодых математиков). Третий автор поддержан Международной стипендиальной инициативой Президента CAS (PIFI, грант 2016VMA078).

PY - 2017/5/1

Y1 - 2017/5/1

N2 - A subgroup H of a group G is pronormal if the subgroups H and Hg are conjugate in 〈H,Hg〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSLn(q), PSUn(q), E6(q), 2E6(q), where in all cases q is odd and n is not a power of 2, and P Sp2n(q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp2n(q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups. The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp2n(q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2m or 2m(22k+1), this group has a nonpronormal subgroup of odd index. If n = 2m, then we show that all subgroups of P Sp2n(q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp2n(q) is still open when n = 2m(22k + 1) and q ≡ ±3 (mod 8).

AB - A subgroup H of a group G is pronormal if the subgroups H and Hg are conjugate in 〈H,Hg〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSLn(q), PSUn(q), E6(q), 2E6(q), where in all cases q is odd and n is not a power of 2, and P Sp2n(q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp2n(q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups. The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp2n(q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2m or 2m(22k+1), this group has a nonpronormal subgroup of odd index. If n = 2m, then we show that all subgroups of P Sp2n(q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp2n(q) is still open when n = 2m(22k + 1) and q ≡ ±3 (mod 8).

KW - finite group

KW - odd index

KW - pronormal subgroup

KW - simple group

KW - symplectic group

UR - http://www.scopus.com/inward/record.url?scp=85021277265&partnerID=8YFLogxK

UR - https://www.elibrary.ru/item.asp?id=31022970

U2 - 10.1134/S0037446617030107

DO - 10.1134/S0037446617030107

M3 - Article

AN - SCOPUS:85021277265

VL - 58

SP - 467

EP - 475

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 3

M1 - 10

ER -