The structure of Bell groups

For any integer n > 1, the variety of n-Bell groups is defined by the law w(x 1, x 2) = [x 1 n , x 2][x 1, x 2 n ]−1. Bell groups were studied by R. Brandl in [2], and by R. Brandl and L.-C. Kappe in [3]. 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.