On $\sigma$-$c$-subnormal subgroups of finite groups

Abstract

Let $ \sigma=\{\sigma_i:i\in I\} $ be a partition of the set $ \mathbb{P} $ of all primes. A finite group $ G $ is called $ \sigma $-primary if the prime divisors, if any, of $|G|$ all belong to the same member of $ \sigma $. A finite group $ G $ is called $ \sigma $-soluble if every chief factor of $ G $ is $ \sigma$-primary. A subgroup $H$ of a group $G$ is called $\sigma$-subnormal in $G$ if there is a chain of subgroups $H=H_0\leq H_1\leq\cdots\leq H_n=G$ such that either $ H_{i-1} $ is normal in $ H_i $ or $ H_{i}/(H_{i-1})_{H_{i}} $ is $ \sigma $-primary for all $ i=1,\dots,n $; A subgroup $H$ of a group $G$ is called $\sigma$-$c$-subnormal in $G$ if there is a subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq H_{\sigma G}$, where the subgroup $H_{\sigma G}$ is generated by all $\sigma$-subnormal subgroups of $G$ contained in $H$. In this paper, we investigate the influence of $\sigma$-$c$-subnormality of some kinds of maximal
 subgroups on $\sigma$-solubility of finite groups, which generalize some known results.