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 $F_r$ be the Free Group of Rank $r$ and let $N$ be the Subgroup generated by the set of $n$th Powers $\{g^n \vert g \in F_r\}$. Then $N$ is a normal subgroup of $F_r$. We define $B(r,n) = F_r/N$ to be the Quotient Group. We call $B(r,n)$ the $r$-generator Burnside group of exponent $n$. It is the largest $r$-generator group of exponent $n$, in the sense that every other such group is a Homeomorphic image of $B(r,n)$. The Burnside problem is usually stated as: ``For which values of $r$ and $n$ is $B(r,n)$ a Finite Group?''

An answer is known for the following values. For $r=1$, $B(1,n)$ is a Cyclic Group of Order $n$. For $n=2$, $B(r,2)$ is an elementary Abelian 2-group of Order $2^r$. For $n=3$, $B(r,3)$ was proved to be finite by Burnside. The Order of the $B(r,3)$ groups was established by Levi and van der Waerden (1933), namely $3^a$ where

$$a \equiv r + {r\choose 2} + {r\choose 3},$$ (1)

where ${n\choose k}$ is a Binomial Coefficient. For $n=4$, $B(r,4)$ 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,

$$\vert B(2,4)\vert = 2^{12}$$ (2)

$$\vert B(3,4)\vert = 2^{69}$$ (3)

$$\vert B(4,4)\vert = 2^{422}$$ (4)

$$\vert B(5,4)\vert = 2^{2728},$$ (5)

while for larger values of $r$ the exact value is not yet known. For $n=6$, $B(r,6)$ was proved to be finite by Hall (1958) with Order $2^a 3^b$, where

$$a \equiv 1 + (r-1) 3^c$$ (6)

$$b \equiv 1 + (r-1) 2^r$$ (7)

$$c \equiv r + {r \choose 2} + {r \choose 3}.$$ (8)

No other Burnside groups are known to be finite. On the other hand, for $r>2$ and $n\geq 665$, with $n$ Odd, $B(r,n)$ is infinite (Novikov and Adjan 1968). There is a similar fact for $r>2$ and $n$ a large Power of 2.

E. Zelmanov was awarded a Fields Medal in 1994 for his solution of the ``restricted'' Burnside problem.

See also Free Group

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 $p$-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.