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 positive integers $r,n$, let $F_r$ be the free group on generators $x_1,\ldots,x_r$ and define\[ B(r,n)=F_r/\langle\!\langle w^n:w\in F_r\rangle\!\rangle, \] where double angle brackets denote normal closure. A group has exponent dividing $n$ if $g^n=1$ for every element $g$. Every group generated by at most $r$ elements with this property is a quotient of $B(r,n)$.

Statement recorded in the supplied source.

Is every finitely generated group of finite exponent finite? Equivalently, is $B(r,n)$ finite for all positive integers $r,n$? This is the historical unrestricted Burnside question; the source discussion records infinite free Burnside groups for large exponents, so the universal assertion has a negative answer. [S2]

Short English statement

Must a group generated by finitely many elements be finite if all its element orders have one common finite bound? The historical universal assertion is false.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3331 (OPG-760). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Burnside problem, node 760, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/burnside_problem.