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A subgroup H of a finite group G is said to be Hall subnormally embedded in G if H is a Hall subgroup of the subnormal closure H S G . In this paper, some new characterizations for finite groups are obtained based on the assumption that some subgroups are Hall subnormally embedded.

Given a finite group G, we define the subgroup D(G) to be the intersection of the normalizers of derived subgroups of all subgroups of G. Set D0=1. Define Di+1(G)/Di(G)=D(G/Di(G)) for i⩾1. By D∞(G) denote the terminal term of the ascending series. It is proved that the derived subgroup G′ is nilpotent if and only if G=D∞(G). Furthermore, if all elements of prime order of G are in D(G), then G is soluble with Fitting length at most 3. In Section 3, it is proved that if the group G satisfies G=D(G), then G′ is nilpotent and G″ has nilpotency class at most 2.

A group G is called an ATI-group if for any abelian subgroup A of G, A ∩ A x = 1 or A for all x ∈ G . In this paper the finite ATI-groups are classified.

The following concept is introduced: a subgroup H of the group G is said to be SS-quasinormal (Supplement-Sylow-quasinormal) in G if H possesses a supplement B such that H permutes with every Sylow subgroup of B. Groups with certain SS-quasinormal subgroups of prime power order are studied. For example, fix a prime divisor p of |G| and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Md(P) denote a family of maximal subgroups P1,…,Pd of P satisfying ⋂i=1d(Pi)=Φ(P), the Frattini subgroup of P. Assume that the group G is p-solvable and every member of some fixed Md(P) is SS-quasinormal in G, then G is p-supersolvable.