For any integer n > 1, the variety of n-Bell groups is defined by the law [x^n,y][x,y^n]^{-1}. Bell groups were studied by R. Brandl, and by R. Brandl and L.-C. Kappe. In this paper we determine the structure of these groups. We prove that if G is an n-Bell group then G/Z_2(G) has finite exponent depending only on n. Moreover, either G/Z_2(G) is locally finite or G has a finitely generated subgroup H such that H=Z(H) is an infinite group of finite exponent. Finally, if G is finitely generated, then the subgroup H may be chosen to be the finite residual of G
We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite p-groups and finite simple groups with the above defined property.
We show that there is a positive constant δ < 1 such that the probability of satisfying either the 2-Engel identity [X1, X2, X2] = 1 or the metabelian identity [[X1, X2], [X3, X4]] = 1 in a finite group is either 1 or at most δ.
A group G is locally graded if every finitely generated nontrivial subgroup of G has a nontrivial finite image. Let N (2, k)* denote the class of groups in which every infinite subset contains a pair of elements that generate a nilpotent subgroup of class at most k. We show that if G is a finitely generated locally graded N (2, it)*-group, then there is a positive integer c depending only on it such that G/Z C (G) is finite.2000 Mathematics subject classification: primary 20F19,20E26.
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