Generalized $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups

Throughout the article, all groups are finite and $G$ always denotes a finite group. Moreover, $\sigma$ is some partition of the set of all primes $\mathbb{P}$, i. e. $\sigma=\{\sigma_i\mid i\in I\}$, where $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$. A $\sigma$-property of a group is any of its properties that do not depend on the choice of the partition $\sigma$ of the set $\mathbb{P}$. This work is devoted to further the study of the $\sigma$-properties of a group. A lot of known results are generalized.