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  "body": "\n## Problem\n\n1. General Burnside problem: let $G$ be a [[finitely generated group|082DWWXMRJ]] in which every element has finite [[order|0WE2NHZJ7B]]. Must $G$ be finite?\n2. Bounded Burnside problem: let $m \\ge 1$ and $n \\ge 1$ be integers. Let $B(m, n)$ be the quotient of the [[free group|0KTW26C80J]] $F_m$ of rank $m$ by the normal subgroup generated by all $w^n$ with $w \\in F_m$. For which $m$ and $n$ is $B(m, n)$ finite?\n\n## Remarks\n\nThe group $B(m, n)$ is the free Burnside group: every group with $m$ generators and [[exponent|0ZDW0H0VX4]] dividing $n$ is a quotient of it.\nSo the bounded problem asks whether all groups with $m$ generators and exponent $n$ are finite of bounded order.\nBurnside posed the problem in 1902.\n\nThe general problem has a negative answer.\nGolod and Shafarevich built infinite finitely generated $p$-groups in 1964, and the [[Grigorchuk group|M0WAT1VQZH]] is a later explicit example.\n\nFor the bounded problem, $B(m, n)$ is finite for $n = 2, 3, 4, 6$ and all $m$.\nFor instance $B(m, 2) \\cong (\\Z/2)^m$, since $x^2 = 1$ for all $x$ forces the group to be abelian.\nNovikov and Adian proved that $B(m, n)$ is infinite for $m \\ge 2$ and all odd $n$ large enough, and later work extended this to large even $n$.\n[[Tarski monster groups|TE19BGFFF1]] are infinite $2$-generated groups of large prime exponent.\nWhether $B(2, 5)$ is finite is open.\n\n## Explanations\n\n### L1 human\n\nIn a finite group, every element returns to the identity after finitely many repetitions.\nBurnside asked about the converse: if a group is generated by finitely many elements and every element returns to the identity, must the group be finite?\nThe answer is no in general, and even if all elements satisfy $x^n = 1$ for one fixed $n$, the answer is no for many large $n$.\nThe question for $2$ generators and $n = 5$ is still open.\n",
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