-semipermutable subgroups of finite groups.
In this note, we study finite groups possessing exactly one nonlinear non-faithful irreducible character. Our main result is to classify solvable groups that satisfy this property. Also, we give examples to show that these groups need not to be solvable in general.
Let ℨ be a complete set of Sylow subgroups of a group G. A subgroup H of G is called ℨ-permutably embedded in G if every Sylow subgroup of H is also a Sylow subgroup of some ℨ-permutable subgroup of G. By using this concept, we obtain some new criteria of p-supersolubility and p-nilpotency of a finite group.