TY - JOUR

T1 - The reduction theorem for relatively maximal subgroups

AU - Guo, Wenbin

AU - Revin, Danila O.

AU - Vdovin, Evgeny P.

N1 - Funding Information: We would like to thank Prof. Andrei Zavarnitsine for his kind help in preparing the manuscript of this article. Wenbin Guo is supported by a NNSF grant of China (Grant No. 11771409) and Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences. Danila O. Revin is supported by Chinese Academy of Sciences President's International Fellowship Initiative, PIFI, (Grant No. 2016VMA078), by RFBR and BRFBR, Project Number 20-51-00007 and by the Programof Fundamental Scientific Research of the SB RAS No. I.1.1., Project Number 0314-2016-0001. Evgeny P. Vdovin is supported by Chinese Academy of Sciences President's International Fellowship Initiative, PIFI, (Grant No. 2017VMA0049) and by the Mathematical Center in Akademgorodok under Agreement No. 075- 15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. Publisher Copyright: © 2022 The Author(s).

PY - 2022/4/1

Y1 - 2022/4/1

N2 - Let be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if A is a normal subgroup of a finite group G then the image of an-maximal subgroup H of G in G/A is not, in general,-maximal in G/A. We say that the reduction-Theorem holds for a finite group A if, for every finite group G that is an extension of A (i.e. contains A as a normal subgroup), the number of conjugacy classes of-maximal subgroups in G and G/A is the same. The reduction-Theorem for A implies that HA/A is-maximal in G/A for every extension G of A and every-maximal subgroup H of G. In this paper, we prove that the reduction-Theorem holds for A if and only if all-maximal subgroups of A are conjugate in A and classify the finite groups with this property in terms of composition factors.

AB - Let be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if A is a normal subgroup of a finite group G then the image of an-maximal subgroup H of G in G/A is not, in general,-maximal in G/A. We say that the reduction-Theorem holds for a finite group A if, for every finite group G that is an extension of A (i.e. contains A as a normal subgroup), the number of conjugacy classes of-maximal subgroups in G and G/A is the same. The reduction-Theorem for A implies that HA/A is-maximal in G/A for every extension G of A and every-maximal subgroup H of G. In this paper, we prove that the reduction-Theorem holds for A if and only if all-maximal subgroups of A are conjugate in A and classify the finite groups with this property in terms of composition factors.

KW - -maximal subgroup

KW - -submaximal subgroup

KW - Complete class

KW - finite simple group

UR - http://www.scopus.com/inward/record.url?scp=85099723993&partnerID=8YFLogxK

U2 - 10.1142/S1664360721500016

DO - 10.1142/S1664360721500016

M3 - Article

AN - SCOPUS:85099723993

VL - 12

JO - Bulletin of Mathematical Sciences

JF - Bulletin of Mathematical Sciences

SN - 1664-3607

IS - 1

M1 - 2150001

ER -