Restricted Burnside problem

Statement

For what values of $n$ is the restricted Burnside group $RB(d,n)$ finite for all $d$? Explicitly, for what values of $n$ is it true that for every natural number $d$, there is a finite group $G$ such that every finite group with exponent dividing $n$ and at most $d$ generators is a quotient group of $G$?

Solution

The restricted Burnside problem has been solved using work by Kostrikin and Zelmanov, and the answer is Yes for all $n$. The conclusion is as follows:

• Reduction of restricted Burnside problem to associated Lie ring is a first step used in all the theorems related to the restricted Burnside problem.
• Hall-Higman theorem on restricted Burnside problem states that the restricted Burnside problem has an answer of Yes for a particular $n$ if and only if it has an answer of Yes for all the maximal prime powers dividing $n$.
• Kostrikin's theorem on restricted Burnside problem states that the answer is Yes for all primes.
• Zelmanov's theorem on restricted Burnside problem states that the answer is Yes for higher prime powers, and hence, combined with the Hall-Higman theorem, gives us that it is always Yes.

Relation with Burnside problem

Note that this problem is related to the Burnside problem in the following sense: a priori, for any $n$ for which the answer to the Burnside problem is Yes, the answer to the restricted Burnside problem must also be Yes. Hence, for any $n$ for which the answer to the restricted Burnside problem is No, the answer to the Burnside problem is also No. However, it is conceivable that there are values of $n$ for which the answer to the Burnside problem is No but the answer to the restricted Burnside problem is Yes.

Of course, a posteriori, since the answer to the restricted Burnside problem is yes for all $n$, this ends up shedding no light on whether the answer to the Burnside problem is yes or no for a particular $n$.