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Proceedings of the Institute of Mathematics of the NAS of Belarus, 2024, Volume 32, Number 1, Pages 31–37
(Mi timb381)
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ALGEBRA AND NUMBER THEORY
On n-multiply σ-locality of a non-empty τ-cloused formation of finite groups
I. N. Safonova Belarusian State University, Minsk, Belarus
Abstract:
All groups under consideration are finite. Let σ={σi∣i∈I} be some partition of the set of all primes, G be a group, σ(G)={σi∣σi∩π(G)≠∅}, F be a class of groups, and σ(F)=⋃G∈Fσ(G).
A function f of the form
f:σ→{formations of groups} is called
a formation σ-function. For any formation σ-function f the class LFσ(f) is defined as follows:
LFσ(f)=(G∣G=1 or G≠1 and G/Oσ′i,σi(G)∈f(σi) for all σi∈σ(G)).
If for some formation σ-function f we have F=LFσ(f), then the class F is called σ-local and f is called a σ-local definition of F.
Every formation is called 0-multiply σ-local. For n⩾1, a formation F is called n-multiply σ-local provided either F=(1) is the class of all identity groups
or F=LFσ(f), where f(σi) is (n−1)-multiply σ-local for all σi∈σ(F).
Let τ(G) be a set of subgroups of G such that
G∈τ(G). Then τ is called a {subgroup functor} if for every epimorphism φ : A→ B and any groups H∈τ(A) and T∈τ(B) we have Hφ∈τ(B) and Tφ−1∈τ(A).
A class of groups F is called
{τ-closed} if τ(G)⊆F for all G∈F.
In this paper, necessary and sufficient conditions for n-multiply σ-locality (n⩾1) of a non-empty τ-closed formation are obtained.
Keywords:
finite group, formations, subgroup functor, σ-local formation, τ-closed formation.
Received: 21.02.2024 Revised: 14.06.2024 Accepted: 18.06.2024
Citation:
I. N. Safonova, “On n-multiply σ-locality of a non-empty τ-cloused formation of finite groups”, Proceedings of the Institute of Mathematics of the NAS of Belarus, 32:1 (2024), 31–37
Linking options:
https://www.mathnet.ru/eng/timb381 https://www.mathnet.ru/eng/timb/v32/i1/p31
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Abstract page: | 66 | Full-text PDF : | 26 | References: | 31 |
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