Characterization of some classes of finite groups

Throughout the paper, all groups are finite and $G$ always denotes a finite group; $\mathbb{P}$ is the set of all primes and $\sigma=\{\sigma_i\mid i\in I\}$ is an arbitrary partition of $\mathbb{P}$. By a $\sigma$-property of a group we mean any property of it that depends on $\sigma$ and that does not imply any restrictions on $\sigma$. In this paper, further applications of the theory of $\sigma$-properties of a group in the study of generalized $T$-groups and other classes of finite groups are analyzed.