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ON THE σ-NILPOTENT NORM AND THE σ-NILPOTENT LENGTH OF A FINITE GROUP

Part of: Representation theory of groups

Published online by Cambridge University Press:  27 February 2020

BIN HU ,
JIANHONG HUANG Open the ORCID record for JIANHONG HUANG [Opens in a new window]  and
ALEXANDER N. SKIBA
BIN HU
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, P.R. China e-mails: hubin118@126.com; jhh320@126.com
JIANHONG HUANG
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, P.R. China e-mails: hubin118@126.com; jhh320@126.com
ALEXANDER N. SKIBA
Affiliation:
Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University, Gomel246019, Belarus e-mail: alexander.skiba49@gmail.com
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Abstract

Let G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes $\Bbb{P}$ . Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use $G^{{\mathfrak{N}}_{\sigma}}$ to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is,

$${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$

Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.

MSC classification

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary: 20D15: Nilpotent groups, $p$-groups 20D30: Series and lattices of subgroups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 121 - 132
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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