Abstract:
In this paper: $G$ is a finite group; $\sigma=\{\sigma_i\mid i\in I\}$ is some partition of the set of all primes $\mathbb{P}$; $\Pi\subseteq\sigma$; $\sigma(n)=\{\sigma_i\mid \sigma_i\cap\pi(n)\ne\varnothing\}$($n$ is an integer) and $\sigma(G)=\sigma(|G|)$.
A group $G$ is said to be: (i) $\sigma$-primary provided $G$ is a $\sigma_i$-group for some $i\in I$; (ii) $\sigma$-nilpotent if $G$ is the direct product of $\sigma$-primary groups; a $\Pi$-group if $\sigma(G)\subseteq\Pi$. A subgroup $A$
of a finte group $G$ is said to be: (i) $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_t=G$ such that either
$A_{i-1}\unlhd A_i$ or $A_i/(A_{i-1})_{A_i}$ is $\sigma$-primary for all $i = 1,\dots, t$; (ii) a Hall $\Pi$-subgroup of $G$ if $A$ is a $\Pi$-group and $\sigma(|G:A|)\cap\Pi=\varnothing$.
We say that a subgroup $H$ of $G$ is strongly $\sigma$-subnormal if $H^G/H_G$ is $\sigma$-nilpotent. In this paper, we prove that the set of all
strongly $\sigma$-subnormal subgroups which permute with a Hall $\Pi$-subgroup of a finite group $G$ forms a sublattice of the lattice of
all subgroups $L(G)$ of $G$.
Keywords:finite group, lattice of subgroups, operator group, sublattice of a lattice, Hall $\Pi$-subgroup.
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