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On n-multiply σ-locality of a non-empty τ-cloused formation of finite groups

Abstract

All groups under consideration are finite. Let $\sigma =\{\sigma_{i} \mid i\in I \}$ be some partition of the set of all primes, $G$ be a group, $\sigma (G)=\{\sigma_i\mid \sigma_i\bigcap \pi (G)\ne \varnothing\} $, $\mathfrak F$ be a class of groups, and $\sigma (\mathfrak{F})=\bigcup_{G\in \mathfrak{F}}\sigma (G).$ A function $f$ of the form $f:\sigma \to\{\text{formations of groups}\}$ is called a formation σ‑function. For any formation σ‑function $f$ the class $LF_{\sigma}(f)$ is defined as follows: $ LF_{\sigma}(f)=(G \mid G=1 \ \text{or }\ G\ne 1\ \text{and }\ G/O_{\sigma_i', \sigma_i}(G) \in f(\sigma_{i}) \ \text{ for all } \sigma_i \in \sigma(G)). $ If for some formation σ‑function $f$ we have $\mathfrak{F}=LF_{\sigma}(f),$ then the class $\mathfrak{F}$ is called $\sigma $-local and $f$ is called a σ‑local definition of $ \mathfrak{F}.$ Every formation is called 0-multiply $\sigma $-local. For $n \geqslant 1,$ a formation $\mathfrak{F}$ is called $n$-multiply $\sigma $-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=LF_{\sigma}(f),$ where $f(\sigma_i)$ is $(n-1)$-multiply σ‑local for all $\sigma_i\in \sigma (\mathfrak{F}).$ Let $\tau(G)$ be a set of subgroups of $G$ such that $G\in \tau(G).$ Then $\tau$ is called a {subgroup functor} if for every epimorphism $\varphi$ : $A \to~B$ and any groups $H \in \tau (A)$ and $T\in \tau (B)$ we have $H^{\varphi}\in\tau(B)$ and $T^{{\varphi}^{-1}}\in\tau(A).$ A class of groups $\mathfrak{F}$ is called $\tau$-closed if $\tau(G)\subseteq\mathfrak{F}$ for all $G\in\mathfrak F.$ In this paper, necessary and sufficient conditions for $n$-multiply σ‑locality $(n\geqslant 1)$ of a non-empty $\tau$-closed formation are obtained.

About the Author

I. N. Safonova
Belarusian State University
Belarus

Minsk



References

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Safonova I.N. 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. 2024;32(1):31-37. (In Russ.)

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