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Let G be a finite group and H a subgroup of G. H is said to be N H -embedded in G if there exists a normal subgroup T of G such that H T is a Hall subgroup of G and H ∩ T ≤ H s ̄ G , where H s ̄ G is the largest s -semipermutable subgroup of G contained in H. In this paper, we give some new characterizations of p -nilpotent and supersolvable groups by using N H -embedded subgroups. Some known results are generalized. [ABSTRACT FROM AUTHOR]

Let G be a finite group, p a prime, P a Sylow p -subgroup of G and d a power of p such that 1 < d < | P |. Let G p ∗ denote the unique smallest normal subgroup of G for which the corresponding factor group is abelian of exponent dividing p − 1. Let 1 , 2 , 3 be classes of all p -groups, p -nilpotent groups and p -supersolvable groups, respectively, G be the -residual of G. Let X ∈ { (G p ∗) 1 , G 2 , G 3 }. A subgroup H of a finite group G is said to have Π -property in G , if for any G -chief factor L / K , | G / K : N G / K ((H ∩ L) K / K) | is a π ((H ∩ L) K / K) -number. A normal subgroup E of G is said to be p -hypercyclically embedded in G if every p - G -chief factor of E is cyclic, where p is a fixed prime. In this paper, we prove that E is p -hypercyclically embedded in G if and only if for some p -subgroups H of E , H ∩ X have Π -property in G. [ABSTRACT FROM AUTHOR]

Let be a finite group. A subgroup of is said to be a Hall -semiembedded subgroup of if is a Hall subgroup of for any , where . In this paper, we investigate the influence of Hall -semiembedded subgroups on the structure of the finite group . Some new results about to be a -group are obtained, where is a saturated formation. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal. [ABSTRACT FROM AUTHOR]