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 PFMT, 2020, Issue 1(42), Pages 65–73 (Mi pfmt698)

MATHEMATICS

Finite groups with generalized $\sigma$-subnormal and $\sigma$-permutable subgroups

N. M. Adarchenko, A. N. Skiba

F. Scorina Gomel State University

Abstract: Throughout this paper, all groups are finite and $G$ always denotes a finite group. We say that a subgroup $H$ of $G$ is nearly modular in $G$ if either $A$ is normal in $G$ or $H_g\ne H^G$ and every chief factor $H/K$ of $G$ between $H_G$ and $H^G$ is nearly central in $G$, that is, $|H/K||GC_G(H/K)|$ divides $pq$ for some primes $p$ and $q$. We say that a subgroup $A$ of $G$ is:
(i) nearly $\sigma$-subnormal in $G$ if $A=\langle L,T\rangle$, where $L$ is a nearly modular subgroup and $T$ is a $\sigma$-subnormal subgroup of $G$;
(ii) nearly $\sigma$-permutable in $G$ if $A=\langle L,T\rangle$, where $L$ is a nearly modular subgroup and $T$ is a $\sigma$-permutable subgroup of $G$.
(iii) weakly $\sigma$-permutable in $G$ if there are a nearly $\sigma$-permutable subgroup $S$ and a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leqslant S\leqslant H$.
In the given paper, we study finite groups with some systems of nearly $\sigma$-subnormal, nearly $\sigma$-permutable and weakly $\sigma$-permutable subgroups. Some known results are generalized.

Keywords: finite group, $n$-maximal subgroup, nearly $\sigma$-subnormal subgroup, nearly $\sigma$-permutable subgroup, $\sigma$-nearly nilpotent group.

Full text: PDF file (435 kB)
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UDC: 512.54
MSC: 20D10, 20D20, 20D30, 20D35
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Citation: N. M. Adarchenko, A. N. Skiba, “Finite groups with generalized $\sigma$-subnormal and $\sigma$-permutable subgroups”, PFMT, 2020, no. 1(42), 65–73

Citation in format AMSBIB
\Bibitem{AdaSki20} \by N.~M.~Adarchenko, A.~N.~Skiba \paper Finite groups with generalized $\sigma$-subnormal and $\sigma$-permutable subgroups \jour PFMT \yr 2020 \issue 1(42) \pages 65--73 \mathnet{http://mi.mathnet.ru/pfmt698}