The restricted Burnside problem, formulated in the 1930s, asks another, related, question:
- If it is known that a group G with m generators and exponent n is finite, can one conclude that the order of G is bounded by some constant depending only on m and n? Equivalently, are there only finitely many finite groups with m generators of exponent n, up to isomorphism?
This variant of the Burnside problem can also be stated in terms of certain universal groups with m generators and exponent n. By basic results of group theory, the intersection of two subgroups of finite index in any group is itself a subgroup of finite index. Let M be the intersection of all subgroups of the free Burnside group B(m, n) which have finite index, then M is a normal subgroup of B(m, n) (otherwise, there exists a subgroup g -1Mg with finite index containing elements not in M). One can therefore define a group B0(m,n) to be the factor group B(m,n)/M. Every finite group of exponent n with m generators is a homomorphic image of B0(m,n). The restricted Burnside problem then asks whether B0(m,n) is a finite group.
In the case of the prime exponent p, this problem was extensively studied by A. I. Kostrikin during the 1950s, prior to the negative solution of the general Burnside problem. His solution, establishing the finiteness of B0(m,p), used a relation with deep questions about identities in Lie algebras in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov, who was awarded the Fields Medal in 1994 for his work.
Read more about this topic: Burnside's Problem
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