Groups finitely presented in Burnside varieties

Abstract

For all sufficiently large odd integers n, the following version of Higman's embedding theorem is proved in the variety Bn of all groups satisfying the identity xn=1. A finitely generated group G from Bn has a presentation G=〈A|R〉 with a finite set of generators A and a recursively enumerable set R of defining relations if and only if it is a subgroup of a group H finitely presented in the variety Bn. It follows that there is a ‘universal’ 2-generated finitely presented in Bn group containing isomorphic copies of all finitely presented in Bn groups as subgroups.