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) |

(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.

© 1996-9

1999-05-26