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 group element 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 group exponent.
Let 
and 
be positive integers, let 
be the free group on 
group
generators, and let 
be the normal subgroup
of 
generated by all 
th
powers 
with 
. The quotient group 
is the 
-generator Burnside group of group
exponent 
.
Every group generated by at most 
elements whose group
exponent divides 
is the image of 
under a surjective group homomorphism. The bounded-exponent Burnside
problem asks for which 
and 
the group 
is a finite group.
An answer is known for the following values. For 
, 
is a cyclic group of
group order 
. For 
, 
is the trivial group.
For 
,
is an elementary
Abelian group of group order 
. For 
, 
was proved to be finite by
Burnside (1902). The group order of the 
groups was established by Levi
and van der Waerden (1933), namely 
where
 |
(1)
|
Here 
is a binomial coefficient. For 
, 
was proved to be finite by
Sanov (1940). For groups of exponent four, the precise
nilpotency class and derived length are known, as are bounds for the group
order. The known exact group orders for 
at 
, 2, ..., 5 are 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 |  | Newman and O'Brien (1996) |
The inequality 
was proved by Burnside (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 (1957, 1958) with group order 
, where
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
These results give positive solutions for every finite number of group generators at group exponents 1, 2, 3, 4, and
6. For 
and odd 
, Novikov and Adjan (1968) proved that 
is infinite. Adian later
reduced the threshold to odd 
(Ivanov 1998, p. 67). Lysenok (1996) proved that
is infinite
for 
and even 
.
The restricted Burnside problem asks whether the group orders of all finite groups generated by at most
elements
with group exponent dividing 
have a bound depending only on 
and 
. The answer is affirmative (Vaughan-Lee 1993). E. Zelmanov
was awarded a Fields medal in 1994 for his solution
(International Mathematical Union 1994).
-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. Jr. "Solution of the Burnside Problem for Exponent
6." Proc. Natl. Acad. Sci. USA 43, 751-753, 1957.