On σ-supersoluble groups and one generalization of CLT-groups.

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes P . A chief factor H / K of G is said to be σ-central (in G ) if the semidirect product ( H / K ) ⋊ ( G / C G ( H / K ) ) is a σ i -group for some i ∈ I . The group G is said to be σ-nilpotent if either G = 1 or every chief factor of G is σ -central. Let G N σ be the σ-nilpotent residual of G , that is, the intersection of all normal subgroups N of G with σ -nilpotent quotient G / N . Then we say that G is σ-supersoluble if each chief factor of G below G N σ is cyclic. In this paper we study properties of σ -supersoluble groups and also consider some applications of such groups in the theory of generalized CLT -groups. [ABSTRACT FROM AUTHOR]