in this paper, we consider the finitely 2-generated groups k(s,l) and g_m as follows:k(s,l) = ;g_m = and find the explicit formulas for the probability of having nth-roots for them. also weinvestigate integers n for which, these groups are n-central.

Abstract In this article, some new sufficient conditions of p -nilpotency finite groups are obtained by using c -normality and Φ-supplementary the maximal or 2-maximal subgroups Sylow -subgroups.

Let G be a finite group, p a fix prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p = 2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.

In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension 10 over any field. It is known that symplectic alternating algebras over GF(3) correspond to a special rich class C of 2-Engel 3-groups of exponent 27 and under this correspondance we will see that the nilpotent algebras corre...