Generalization of some classical results of Srinivasan

Abstract

Let $\sigma =\{\sigma _i:i\in I\}$ be a partition of the set ${\mathbb {P}}$ of all prime numbers, G a finite group, $\pi (G)$ the set of all primes dividing |G|, and $\sigma (G)=\{\sigma _i : \sigma _i\cap \pi (G)\ne \emptyset \}$. The group G is called a $\sigma$-group if G has a set of subgroups ${\mathcal {H}}$ such that every non-trivial subgroup contained in ${\mathcal {H}}$ is a Hall $\sigma _i$-subgroup of G and ${\mathcal {H}}$ contains exactly one Hall $\sigma _i$-subgroup of G for every $\sigma _i\in \sigma (G).$ In this paper, we investigate the structure of the $\sigma$-groups G by using the $\sigma$-normality, $\sigma$-permutability, and $\sigma$-subnormality of maximal subgroups of elements in ${\mathcal {H}}$. Some criteria of supersolubility and $\sigma$-solubility of G are obtained, which generalize some classical results of Srinivasan.