Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called Hall normally embedded in [Formula: see text] if [Formula: see text] is a Hall subgroup of the normal closure [Formula: see text]. In this paper, we investigate the structure of a finite group [Formula: see text] under the assumption that certain subgroups of prime power order are Hall normally embedded in [Formula: see text].

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be a BNA-subgroup of [Formula: see text] if either [Formula: see text] or [Formula: see text] for all [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be a weakly BNA-subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a BNA-subgroup of [Formula: see text]. In this paper, we investigate the structure of a finite group [Formula: see text] under the assumption that every minimal subgroup of [Formula: see text] not having a supersolvable supplement in [Formula: see text] is a weakly BNA-subgroup of [Formula: see text].

In this paper, we establish the existence of nonoscillatory solutions to the neutral dynamic equation x(t)-b a p(t, η)x(g(t, η)) η + d c ω(t, ν)x(h(t, ν)) ν = 0 on a time scale T. Some examples are given to illustrate the main results.