Burnside Problem

A problem originating with W. Burnside (1902), who wrote, A still undecided point in the theory of discontinuous groups is whether the Order of a Group may be not finite, while the order of every operation it contains is finite.'' This question would now be phrased as Can a finitely generated group be infinite while every element in the group has finite order?'' (Vaughan-Lee 1990). This question was answered by Golod (1964) when he constructed finitely generated infinite p-Group. These Groups, however, do not have a finite exponent.

Let be the Free Group of Rank and let be the Subgroup generated by the set of th Powers . Then is a normal subgroup of . We 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 Homeomorphic 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 Order . For , is an elementary Abelian 2-group of Order . For , was proved to be finite by Burnside. The Order of the groups was established by Levi and van der Waerden (1933), namely where

 (1)

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 Order. For example,
 (2) (3) (4) (5)

while for larger values of the exact value is not yet known. For , was proved to be finite by Hall (1958) with Order , where
 (6) (7) (8)

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.

References

Burnside, W. On an Unsettled Question in the Theory of Discontinuous Groups.'' Quart. J. Pure Appl. Math. 33, 230-238, 1902.

Golod, E. S. On Nil-Algebras and Residually Finite -Groups.'' Isv. Akad. Nauk SSSR Ser. Mat. 28, 273-276, 1964.

Hall, M. Solution of the Burnside Problem for Exponent Six.'' Ill. J. Math. 2, 764-786, 1958.

Levi, F. and van der Waerden, B. L. Über eine besondere Klasse von Gruppen.'' Abh. Math. Sem. Univ. Hamburg 9, 154-158, 1933.

Novikov, P. S. and Adjan, S. I. Infinite Periodic Groups I, II, III.'' Izv. Akad. Nauk SSSR Ser. Mat. 32, 212-244, 251-524, and 709-731, 1968.

Sanov, I. N. Solution of Burnside's problem for exponent four.'' Leningrad State Univ. Ann. Math. Ser. 10, 166-170, 1940.

Vaughan-Lee, M. The Restricted Burnside Problem, 2nd ed. New York: Clarendon Press, 1993.