We show that the class of groups whichhave monoid presentations by means of finite special[λ]-confluent string-rewriting systems strictly contains the class of plain groups(the groups which are free products of a finitely generated freegroup and finitely many finite groups),and that any groupwhich has an infinite cyclic central subgroupcan be presented by such a string-rewriting system if and only if it is thedirect product of an infinite cyclic group and a finite cyclic group.