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.
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