2 Groups

For basic definitions and results of group theory please see a standard text such as [2] or [8].

In a finite group G every element g satisfies gⁿ = e for some least positive integer n called the order of g and denoted by o(g). This leads us to,

Problem 1 (General Burnside Problem)   Let G be a finitely generated group such that for every element g of G there is a positive integer N_g so that g^N_g = e. Then is G finite?

When G arises as a group of n×n matrices (or more formally when G is a linear group) it was shown by Burnside that the answer is yes (a simple proof is outlined in Appendix A). However, in 1964 Golod and Shafarevich [3] showed that this is not true for all groups. Thereafter, Alyoshin [1], Suschansky [12], Grigorchuk [4] and Gupta-Sidki [5] gave various counter-examples.

We can tighten the above conjecture since we know that, o(g) divides o(G) the number of elements of the set of elements of G. Thus we can formulate,

Problem 2 (Ordinary Burnside Problem)   Let G be a finitely generated group for which there is a positive integer N such that for every element g we have g^N = e. Then is G finite?

(We call the smallest such integer N the exponent of G.) In 1968 Novikov and Adian [10] gave counter-examples for groups of odd exponents for the Ordinary Burnside Problem.

If we are primarily interested only in finite groups and their classification then we can again restrict the problem further. Thus Magnus [9] formulated the following problem.

Problem 3 (Restricted Burnside Problem)   Is there a finite number A(k, N) of finite groups which are generated by k elements and have exponent N?

Alternatively one can ask if the order of all such groups is uniformly bounded. Hall and Higman [6] proved that the Restricted Burnside problem for a number N which can be factorised as p₁^{n₁ ...} p_r^n_r follows from the following three hypothesis:

1. The Restricted Burnside Problem is true for p_i^n_i.
2. There are at most finitely many finite simple group quotients which are k-generated and have exponent N.
3. For each finite simple group quotient G as above the group of outer automorphisms of G is a solvable group.

Thus modulo the latter two problems which have to do with the Classification of Finite Simple groups, we reduced to a study of the Restricted Burnside Problem for p-groups. We note that a key step in the Classification of Finite Simple groups was the celebrated theorem of Feit and Thompson which won a Fields Medal in 1970. According to this theorem if N is odd then there are no finite simple groups in items (2) and (3) above.