Characterizations on Solvable or p-Supersolvable Groups

Compared to previous research, we provide a new perspective to understand the relationship between p -local conditions and the structure of a group, i.e., for a meet-irreducible subgroup, we choose a Sylow p -subgroup of its trace. Further, for the chosen primary subgroups, we investigate the influence of its complementarity or s -semipermutability on the structure of a group and obtain some new sufficient and necessary conditions of solvability, p -supersolvability or p -nilpotency. To some extent, the main results can be regarded as an generalization of Frobenius’s theorem on p -nilpotency or Hall’s criterion for solvability, respectively.