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Abstract: In several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named -groups. Groups with all subgroups being -groups are called groups with the basis property. In the first part of this paper we correct and generalize some characterizations of groups with the basis property. In the second part we consider groups satisfying analogous conditions, but only for sets of generators of prime power orders. The class of groups introduced here is much larger then the class of groups with the basis property. For example, it contains all nilpotent groups. [Copyright &y& Elsevier]

Abstract: Let be a Fitting class and a formation. We call a subgroup of a finite group G the – -norm of G if is the intersection of the normalizers of the products of the -residuals of all subgroups of G and the -radical of G. Let π denote a set of primes and let denote the class of all finite π-groups. We call the subgroup of G the -norm of G. A normal subgroup N of G is called -hypercentral in G if either or and every G-chief factor below N of order divisible by at least one prime in π is -central in G. Let denote the -hypercentre of G, that is, the product of all -hypercentral normal subgroups of G. In this paper, we study the properties of the – -norm, especially of the -norm of a finite group G. In particular, we investigate the relationship between the -norm and the -hypercentre of G. [Copyright &y& Elsevier]

Abstract: R. Baer and Wielandt in 1934 and 1958, respectively, considered the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this paper, for a finite group G, we define the subgroup to be the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Set . Define for . By denote the terminal term of the ascending series. It is proved that if and only if the nilpotent residual is nilpotent. Furthermore, if all elements of prime order of G are in , then G is solvable and , where is p-length of G for , where means the set of prime divisors of . [Copyright &y& Elsevier]

Abstract: In this paper, we study the finite 2-group G such that for every . We prove that , is abelian and . We also get that . [Copyright &y& Elsevier]