A characterization of finite $\sigma$-soluble $P\sigma T$-groups

Abstract

Let \mathfrak{F} be a non-empty class of groups, G a finite group and \mathcal{L}(G) be the lattice of all subgroups of G . A chief factor H/K of G is \mathfrak{F} -central in G if (H/K)\rtimes (G/C_G(H/K))\in \mathfrak{F} . Let \mathcal{L}_{c\mathfrak{F}}(G) be the set of subgroups A of G such that every chief factor of G between A^G and A_G is \mathfrak{F} -central in G ; let \mathcal{L}_{\mathfrak{F}}(G) be the set of subgroups A of G such that A^G/A_G\in \mathfrak{F} . In this paper, we study the influence of \mathcal{L}_{\mathfrak{F}}(G) and \mathcal{L}_{c\mathfrak{F}}(G) on the structure of G , where \mathfrak{F} is a normally hereditary saturated formation containing all \sigma -nilpotent groups and D=G^{\mathfrak{F}} is \sigma -soluble. Moreover, we give a new characterization of a finite \sigma -soluble group to be a P\sigma T -group.