On $\sigma$-properties of finite groups III

Let $G$ be a finite group and $\sigma=\{\sigma_i\mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$. Let $\Pi\subseteq\sigma$. We say that a subgroup $A$ of $G$ is $\Pi$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_t=G$ such that either $A_{i-1}$ is normal in $A_i$ or $A_i/(A_{i-1})_{A_i}$ is a $\sigma_j$-group for some $\sigma_j\in\Pi$ for all $i=1,\dots,t$. In this paper, we discuss properties of $\Pi$-subnormal subgroups and some other $\sigma$-properties of finite groups. The work continues the research in the papers [1]–[5].