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When is the search of relatively maximal subgroups reduced to quotient groups?
Wen Bin Guoab,
D. O. Revincde a School of Science, Hainan University, Haikou, Hainan, P. R. China
b Department of Mathematics, University of Science and Technology of China, Hefei, P. R. China
c Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
d N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
e Novosibirsk State University
Abstract:
Let
$\mathfrak{X}$ be a class finite groups closed under taking subgroups, homomorphic images, and extensions, and
let
$\mathrm{k}_{\mathfrak{X}}(G)$ be the number of conjugacy classes
$\mathfrak{X}$-maximal subgroups of a finite group
$G$.
The natural problem calling for a description, up to conjugacy, of
the
$\mathfrak{X}$-maximal subgroups of a given finite group is not inductive.
In particular, generally speaking, the image of an
$\mathfrak{X}$-maximal
subgroup is not
$\mathfrak{X}$-maximal in the image of a homomorphism.
Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal
$\mathfrak{X}$-subgroups (for example, the homomorphisms whose kernels are
$\mathfrak{X}$-groups).
Under such homomorphisms, the image of an
$\mathfrak{X}$-maximal subgroup is always
$\mathfrak{X}$-maximal,
and, moreover, there is a natural bijection between the conjugacy classes
of
$\mathfrak{X}$-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
$\phi$
from a group
$G$,
the equality $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$
holds if and only if
$\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$,
which in turn is equivalent to the fact that the composition factors of the kernel of
$\phi$ lie in an explicitly given list.
Keywords:
finite group, complete class, $\mathfrak{X}$-maximal subgroup, Hall subgroup, reduction $\mathfrak{X}$-theorem.
UDC:
512.542
MSC: 20F28,
20D06,
20E22 Received: 29.10.2021
Revised: 30.01.2022
DOI:
10.4213/im9277
Fulltext:
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