On the centralizer of the $\sigma$-nilpotent residual of the $\sigma$-subnormal subgroup

Throughout this paper, all groups are finite and $G$ always denotes a finite group. Moreover, $\sigma$ is some partition of the set of all primes $\mathbb{P}$, that is, $\sigma=\{\sigma_i\mid i\in I\}$, where $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$. The group $G$ is said to be: $\sigma$-primary if $G$ is a $\sigma_i$-group for some $i$; $\sigma$-nilpotent if every chief factor $H/K$ of $G$ is $\sigma$-central in $G$, that is, $(H/K)\rtimes(G/C_G(H/K))$ is $\sigma$-primary. The symbol $G^{\mathfrak{N}_\sigma}$ denotes the $\sigma$-nilpotent residual of $G$, that is, the intersection of all normal subgroups $N$ of $G$ such that $G/N$ is $\sigma$-nilpotent; $Z_\sigma(G)$ is the $\sigma$-nilpotent hypercentre of $G$, that is, the product of all normal subgroups $N$ of $G$ such that either $N=1$ of $N\ne1$ and every chief factor of $G$ below $N$ is $\sigma$-central in $G$. A subgroup $A$ of $G$ is said to be $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_n=G$ such that either $A_{i-1}\unlhd A_i$ or $A_i/(A_{i-1})_{A_i}$ is $\sigma$-primary for all $i=1,\dots,n$. In this paper, we prove that if $S$ be a $\sigma$-subnormal subgroup of $G$ and $Z_\sigma(E)=1$ for every subgroup $E$ of $G$ such that $S\leqslant E$, then $C_G(S^{\mathfrak{N}_\sigma})\leqslant S^{\mathfrak{N}_\sigma}$.