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

(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 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).

reference

1

2

Tobin (1954)

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

(2)

(3)

(4)

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.

Bayes, A. J.; Kautsky, J.; and Wamsley, J. W. "Computation in Nilpotent Groups (Application)." In Proceedings of the
Second International Conference on the Theory of Groups. Held at the Australian National
University, Canberra, August 13-24, 1973 (Ed. M. F. Newman). New
York: Springer-Verlag, pp. 82-89, 1974.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.Gupta, N. D. and Newman, M. F. "The Nilpotency
Class of Finitely Generated Groups of Exponent Four." In Proceedings of the
Second International Conference on the Theory of Groups. Held at the Australian National
University, Canberra, August 13-24, 1973 (Ed. M. F. Newman). New
York: Springer-Verlag, pp. 330-332, 1974.Hall, M. "Solution
of the Burnside Problem for Exponent Six." Ill. J. Math.2, 764-786,
1958.Havas, G. and Newman, M. F. "Application of Computers
to Questions Like Those of Burnside." In Burnside
Groups. Proceedings of a Workshop held at the University of Bielefeld, Bielefeld,
June-July 1977. New York: Springer-Verlag, pp. 211-230, 1980.Levi,
F. and van der Waerden, B. L. "Über eine besondere Klasse von Gruppen."
Abh. Math. Sem. Univ. Hamburg9, 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.O'Brien,
E. and Newman, M. F. "Application of Computers to Questions Like Those
of Burnside, II." Internat. J. Algebra Comput.6, 593-605, 1996.Sanov,
I. N. "Solution of Burnside's problem for exponent four." Leningrad
State Univ. Ann. Math. Ser.10, 166-170, 1940.Sloane, N. J. A.
Sequences A079682 and A116398
in "The On-Line Encyclopedia of Integer Sequences."Tobin,
J. J. On Groups with Exponent 4. Thesis. Manchester, England: University
of Manchester, 1954.Vaughan-Lee, M. The
Restricted Burnside Problem, 2nd ed. New York: Clarendon Press, 1993.