Groups with identities

Abstract

We characterize the soluble groups which admit a disjunction of monoidal identities. This notion has been introduced by M. Boffa and is equivalent to admitting the elimination of inverses. We use the result of Rosenblatt that a finitely generated soluble group which does not contain the free monoid on two generators is quasi-nilpotent. We also obtain partial results for the class of elementary amenable groups and its subclass of locally finite groups.