Finite Cyclic Coverings And Verbal Subgroups

In this paper, We studied finite cyclic coverings and the property of group word. Firstly we proved that when w= [ak,bk] and the verbal subset ?{G} can be covered by finitely many cyclic subgroups, then the corresponding verbal subgroup ?(G) is either finite or cyclic. Secondly, we proved that when ?= [ak, bl] and the verbal subset ?{G} can be covered by finitely many cyclic subgroups. If G is finitely generated and G does not have any elements like [xk,yl] of finite order, apart from 1, then the corresponding verbal subgroup ?{G) is cyclic-by-finite. Finally, If we further assume that G is finite-by-nilpotent, then the corresponding verbal subgroup ?(G) is either cyclic or finite.