Restricted Burnside problem

Remarks

Kostrikin proved the case where $n$ is prime. Hall and Higman reduced the general case to prime powers $n = p^k$. Their reduction assumes the Schreier conjecture on outer automorphism groups of finite simple groups, which follows from the classification of finite simple groups. Zelmanov solved the prime power case around 1990, for which he received a Fields Medal. The proofs pass to a Lie ring built from a central series of the group, which satisfies Engel identities. They show that such Lie rings are nilpotent under the right finiteness conditions.

Example: for $n = 2$, every group of exponent $2$ is abelian, and $R(m, 2) = (\Z/2)^m$ has order $2^m$.

The 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.

Going deeper

For a student

The theorem gives a largest finite group $R(m, n)$ with $m$ generators and exponent dividing $n$, and all others are its quotients. It is a statement about finite groups, but its proof goes through Lie rings. Kostrikin and Zelmanov proved that Lie algebras satisfying an Engel condition are locally nilpotent under suitable hypotheses. It contrasts with the bounded Burnside problem, where the free Burnside group can be infinite while all its finite quotients have bounded order.