Subgroups of groups finitely presented in Burnside varieties

TL;DR

The paper proves a version of Higman's embedding theorem for groups in Burnside varieties with large odd exponents, showing that certain finitely generated groups can be embedded into finitely presented groups within the same variety.

Contribution

It establishes a Higman-type embedding theorem for Burnside varieties with large odd exponents, demonstrating the existence of a universal finitely presented group in this setting.

Findings

Finitely generated groups in ${ m Burnside}_n$ embed into finitely presented groups in the same variety.

Existence of a universal 2-generated finitely presented group in ${ m Burnside}_n$ containing all finitely presented groups as subgroups.

Extension of Higman's embedding theorem to Burnside varieties with large odd exponents.

Abstract

For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\cal B}_n$ has a presentation $G=\langle A\mid R\rangle$ 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 ${\cal B}_n$. It follows that there is a 'universal' $2$-generated finitely presented in ${\cal B}_n$ group containing isomorphic copies of all finitely presented in ${\cal B}_n$ groups as subgroups.