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Huppert and Manz have determined the nonsolvable groups whose character degrees are products of at most two prime numbers. In this paper, we change the condition from “degrees of a group are products of at most two prime divisors” to “degrees of all proper groups of a group are products of at most two prime divisors” and determine the structure of finite groups with such condition.

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be a partial [Formula: see text]-subgroup of [Formula: see text] if there exists a chief series [Formula: see text] of [Formula: see text] such that [Formula: see text] either covers or avoids each non-Frattini chief factor of [Formula: see text]. In this paper, we study the influence of the partial [Formula: see text]-subgroups on the structure of finite groups. Some new characterizations of the hypercyclically embedded subgroups, [Formula: see text]-nilpotency and supersolubility of finite groups are obtained.

Assume that a finite group G admits a Frobenius group of automorphisms F H with kernel F and complement H such that C G (F ) = 1. In this paper, we investigate this situation and prove that if C G (H) is supersoluble and C G ′ (H) is nilpotent, then G is supersoluble. Also, we show that G is a Sylow tower group of a certain type if C G (H) is a Sylow tower group of the same type.