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  "body": "\n## Statement\n\nLet $m \\ge 1$ and $n \\ge 1$ be integers.\nThen there are only finitely many finite groups, up to isomorphism, that are generated by $m$ elements and have [[exponent|0ZDW0H0VX4]] dividing $n$.\nEquivalently, the free Burnside group $B(m, n)$ has only finitely many [[normal subgroups|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]].\nThen the quotient $R(m, n)$ of $B(m, n)$ by their intersection is finite.\nEvery finite $m$-generated group of exponent dividing $n$ is a quotient of it.\n\n## Remarks\n\nKostrikin proved the case where $n$ is prime.\nHall and Higman reduced the general case to prime powers $n = p^k$.\nTheir reduction assumes the Schreier conjecture on outer automorphism groups of finite simple groups, which follows from the classification of finite simple groups.\nZelmanov solved the prime power case around 1990, for which he received a Fields Medal.\nThe proofs pass to a Lie ring built from a central series of the group, which satisfies Engel identities.\nThey show that such Lie rings are nilpotent under the right finiteness conditions.\n\nExample: for $n = 2$, every group of exponent $2$ is abelian, and $R(m, 2) = (\\Z/2)^m$ has order $2^m$.\n\nThe restriction to finite groups is necessary: the group $B(m, n)$ itself is infinite for $m \\ge 2$ and many large $n$, by the negative answer to the bounded [[Burnside problem|XWEMJ7T2XV]].\n\n## Explanations\n\n### L1 human\n\nThe Burnside problem asks whether a group with finitely many generators, in which $x^n = 1$ for every $x$, must be finite.\nThe answer is often no.\nThe restricted version only looks at the finite groups of this kind.\nIt says that they cannot be arbitrarily large: the number of generators and the exponent bound their order.\n\n### L2 human\n\nThe theorem gives a largest finite group $R(m, n)$ with $m$ generators and exponent dividing $n$, and all others are its quotients.\nIt is a statement about finite groups, but its proof goes through Lie rings.\nKostrikin and Zelmanov proved that Lie algebras satisfying an Engel condition are locally nilpotent under suitable hypotheses.\nIt contrasts with the bounded [[Burnside problem|XWEMJ7T2XV]], where the free Burnside group can be infinite while all its finite quotients have bounded order.\n",
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