A subgroup H of a finite group G is said to be “semi-cover-avoiding in G” if there is a chief series of G such that H covers or avoids every chief factor of the chief series. In this article, some new characterizations for finite solvable groups are obtained based on the assumption that some subgroups have semi-cover-avoiding properties in the groups.

A subgroup H of a finite group G is said to be c–supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is contained in core_G (H). In this paper some results for finite p–nilpotent groups are given based on some subgroups of Pc–supplemented in G, where p is a prime factor of the order of G and P is a Sylow p–subgroup of G. We also give some applications of these results.