On the laws of a metabelian variety

INTR~D~JCTI~NMany varieties of groups are known to have a finite basis for their laws (for the definition and general properties of varieties, see [6]-[8]).Among such varieties are any variety consisting of nilpotent groups of bounded class [4] and any variety generated by a finite group [9].It is not known whether all varieties have a finite basis for their laws.In this paper we prove the following theorem.THEOREM A.Any variety consisting of metabelian groups has a$nite basis for its laws.Because of the relationship between varieties of groups and verbal subgroups of free groups, Theorem A is equivalent to THEOREM B. The free metabelian group of countable rank has maximum condition on verbal subgroups.Our main result is stronger than this.THEOREM C.