Bounded Burnside Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $m\ge1$, let $F_m$ be the free group on $m$ generators. For $n\ge1$, let $N_{m,n}$ be the normal subgroup generated by $\{w^n:w\in F_m\}$, and put \[ B(m,n)=F_m/N_{m,n}. \] This is the free $m$-generator group satisfying $g^n=1$ for every element $g$; every $m$-generated group satisfying that identity is a quotient. Finiteness here concerns this entire group, rather than just its finite quotients.

Statement recorded in the supplied source.

Determine all pairs $(m,n)\in\mathbb N^2$ for which $B(m,n)$ is finite. In particular, is the free two-generator group of exponent five, $B(2,5)$, finite? [S2]

Short English statement

Which numbers of generators and exponents give a finite free Burnside group? The question includes, specifically, the exponent-five group on two generators.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1109 (ALG-009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Miguel Donoso-Echenique and Eduardo Silva, Free Burnside groups of large odd exponent have cost 1, arXiv:2608.20472 (2026), Section 1. https://arxiv.org/abs/2608.20472.