“Burnside Problem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Burnside Problem

释义

Burnside Problem

A problem originating with W. Burnside (1902), who wrote, ``A still undecided point in the theory of discontinuous groupsis whether the Order of a Group may be not finite, while the order of every operation itcontains is finite.'' This question would now be phrased as ``Can a finitely generated group be infinite while everyelement in the group has finite order?'' (Vaughan-Lee 1990). This question was answered by Golod (1964) when he constructedfinitely 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 ofth Powers . Then is a normal subgroup of . We define to bethe Quotient Group. We call the -generator Burnside group of exponent . It is the largest -generatorgroup of exponent , in the sense that every other such group is a Homeomorphic image of . The Burnside problemis 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 establishedby Levi and van der Waerden (1933), namely where

(1)

where

is a Binomial Coefficient. For

was proved to be finite by Sanov (1940). Groups ofexponent four turn out to be the most complicated for which a Positive solution is known. The precise nilpotency class andderived 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,

isinfinite (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.

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 -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.

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