TY - JOUR

T1 - When is the search of relatively maximal subgroups reduced to quotient groups?

AU - Вэньбинь, Го

AU - Ревин, Данила Олегович

N1 - W. Guo was supported by the National Natural Science Foundation of China (project no. 12171126), Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences and Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province. D. O. Revin was supported by RFBR and BRFBR, project no. 20-51-00007 and by the Russian Ministry for Education and Science within the framework of the state commission for the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (project no. FWNF-2022-0002).

PY - 2022

Y1 - 2022

N2 - Let X be a class finite groups closed under taking subgroups, homomorphic images, and extensions, and let kX(G) be the number of conjugacy classes X-maximal subgroups of a finite group G. The natural problem calling for a description, up to conjugacy, of the X-maximal subgroups of a given finite group is not inductive. In particular, generally speaking, the image of an X-maximal subgroup is not X-maximal in the image of a homomorphism. Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal X-subgroups (for example, the homomorphisms whose kernels are X-groups). Under such homomorphisms, the image of an X-maximal subgroup is always X-maximal, and, moreover, there is a natural bijection between the conjugacy classes of X-maximal subgroups of the image and preimage. In the present paper, all such homomorphisms are completely described. More precisely, it is shown that, for a homomorphism ϕ from a group G, the equality kX(G)=kX(imϕ) holds if and only if kX(kerϕ)=1, which in turn is equivalent to the fact that the composition factors of the kernel of ϕ lie in an explicitly given list.

AB - Let X be a class finite groups closed under taking subgroups, homomorphic images, and extensions, and let kX(G) be the number of conjugacy classes X-maximal subgroups of a finite group G. The natural problem calling for a description, up to conjugacy, of the X-maximal subgroups of a given finite group is not inductive. In particular, generally speaking, the image of an X-maximal subgroup is not X-maximal in the image of a homomorphism. Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal X-subgroups (for example, the homomorphisms whose kernels are X-groups). Under such homomorphisms, the image of an X-maximal subgroup is always X-maximal, and, moreover, there is a natural bijection between the conjugacy classes of X-maximal subgroups of the image and preimage. In the present paper, all such homomorphisms are completely described. More precisely, it is shown that, for a homomorphism ϕ from a group G, the equality kX(G)=kX(imϕ) holds if and only if kX(kerϕ)=1, which in turn is equivalent to the fact that the composition factors of the kernel of ϕ lie in an explicitly given list.

KW - finite group

KW - complete class

KW - X-maximal subgroup

KW - Hall subgroup

KW - reduction X-theorem

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85165393590&origin=inward&txGid=9be346b9a93b8d4b71e6838695453dfa

UR - https://www.mendeley.com/catalogue/9e5f3c9d-3001-3268-828a-82613bd8c7ea/

U2 - 10.4213/im9277e

DO - 10.4213/im9277e

M3 - Article

VL - 86

SP - 1102

EP - 1122

JO - Izvestiya Mathematics

JF - Izvestiya Mathematics

SN - 1064-5632

IS - 6

M1 - 3

ER -