Two Theorems about Equationally Noetherian Groups

AbstractAn algebraic set over a groupGis the set of all solutions of some system {f(x1,…,xn)=1∣f∈G∗〈x1,…,xn〉} of equations overG. A groupGis equationally noetherian if every algebraic set overGis the set of all solutions of a finite subsystem of the given one. We prove that a virtually equationally noetherian group is equationally noetherian and that the quotient of an equationally noetherian group by a normal subgroup which is a finite union of algebraic sets is again equationally noetherian. On the other hand, the wreath productW=U≀Tof a non-abelian groupUand an infinite groupTis not equationally noetherian