Let K be a commutative ring with identity and N the free nilpotent K-algebra on a nonempty set X. Then N is a group with respect to the circle composition. We prove that the subgroup generated by X is relatively free in a suitable class of groups, depending on the choice of K. Moreover, we get unique representations of the elements in terms of basic commutators. In particular, if K is of characteristic 0 the subgroup generated by X is freely generated by X as a nilpotent group.

We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.

In this paper we investigate the special automata over nite rank free groups and estimate asymptotic characteristics of sets they accept. We show how one can decompose an arbitrary regular subset of a nite rank free group into disjoint union of sets accepted by special automata or special monoids. These automata allow us to compute explicitly generating functions, l- measures and Cesaro measure of thick monoids. Also we improve the asymptotic classification of regular subsets in free groups.