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

Let G be a finite group and sigma = {sigma(i)\i is an element of I} be a partition of the set of all primes P. A chief factor H/K of C is said to sigma-central (in G) if the semidirect product (H/K) (sic) (G/C-G (H/K)) is a sigma(i)-group for some i is an element of I. The group G is said to be sigma-nilpotent if either G = 1 or every chief factor of G is sigma-central. Let Gn(sigma) be the sigma-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with sigma-nilpotent quotient G/N. Then we say that G is sigma-supersoluble if each chief factor of G below Gn(sigma) is cyclic. In this paper we study properties of sigma-supersoluble groups and also consider some applications of such groups in the theory of generalized CLT-groups. (C) 2018 Elsevier Inc. All rights reserved.