The Burnside problem originated with Burnside (1902), who wrote, "A still undecided point in the theory of discontinuous groups is whether the group order of a group may be not finite, while the order of every operation it contains is finite." This question would now be phrased, "Can a finitely generated group be infinite while every element in the group has finite order?" (Vaughan-Lee 1993). This question was answered by Golod (1964) when he constructed finitely generated infinite p-groups. These groups, however, do not have a finite exponent.
Let be the free group of group rank and let be the normal subgroup generated by the set of th powers . Then is a normal subgroup of . Define to be the quotient group. We call the -generator Burnside group of exponent . It is the largest -generator group of exponent , in the sense that every other such group is a homomorphic image of . The Burnside problem is usually stated as, "For which values of and is a finite group?"
An answer is known for the following values. For , is a cyclic group of group order . For , is an elementary Abelian 2-group of group order . For , was proved to be finite by Burnside. The group order of the groups was established by Levi and van der Waerden (1933), namely where
where is a binomial coefficient. For , was proved to be finite by Sanov (1940). Groups of exponent four turn out to be the most complicated for which a positive solution is known. The precise nilpotency class and derived length are known, as are bounds for the group order, as summarized in the following table. The first few values for , 2, ... are 4, 4096, 590295810358705651712, ... (OEIS A079682), corresponding to 2 to the powers 2, 12, 69, 422, 2728, ... (OEIS A116398).
|3||Bayes et al. (1974)|
|4||Havas and Newman (1980)|
|5||O'Brien and Newman (1996)|
The inequality was proved by Burnside in 1902, who also claimed equality. The result was proved with help from a computer after the inequality had been obtained "by hand" by Gupta and Newman (1974).
For larger values of the exact value is not yet known. For , was proved to be finite by Hall (1958) with group order , where
No other Burnside groups are known to be finite. On the other hand, for and , with odd, is infinite (Novikov and Adjan 1968). There is a similar fact for and a large power of 2.
E. Zelmanov was awarded a fields medal in 1994 for his solution of the "restricted" Burnside problem.