On the Burnside Problem on Periodic Groups

It is proved that the free m-generated Burnside groups B(m, n) of exponent n are infinite provided that m > 1 , n > 248 . In 1902 William Burnside posed the following problem [2]. Does a group G have to be finite provided that G has a finite set of generators and its elements satisfy the identity x" — 1 ? In other words, must a finitely generated group G of exponent n be finite? In the same paper, Burnside proved that the problem was solved in the affirmative for groups of exponents 2, 3 and for 2-generated groups of exponent 4 as well. In 1940 Sanov [ 12] obtained a positive solution to the Burnside problem for the case of exponent 4. The next significant step was made by Marshall Hall [4] in 1957 when he solved the problem in the affirmative for the exponent of 6. In 1964 Golod [3] found the first example of an infinite periodic group with a finite number of generators. Although that example did not satisfy the identity x" = 1, i.e., the group was of unbounded exponent, it gave the first positive evidence that the Burnside problem might not be solved affirmatively for all exponents (and it might possibly fail for very large exponents). In 1968 Novikov and Adian achieved a real breakthrough in a series of fundamental papers [9] in which some ideas put forward by Novikov [8] in 1959 were developed to prove that there are infinite periodic groups of odd exponents « > 4381 with m > 1 generators. Later, Adian [1] improved the estimate up to n > 665 (n is odd again). Notice in the papers [9] that, in fact, the free Burnside groups B(m, n) = Fm/F£,, where Fm is a free group of rank m > 1 and F£, is the normal subgroup of ¥m generated by all nth powers (with odd n > 4381) of elements of Fm, were constructed and studied. Using a very complicated inductive construction, Novikov and Adian presented the group B(m, n) by defining relations of the form A" — 1, where A's are some specially chosen elements of Fw , and studied their consequences. They not only obtained the result that the group B(/n, n) is infinite but also other important information about B(m, n). For example, it was proved that the word and conjugacy problems are solvable in B(m, n) and that any finite or abelian Received by the editors January 7, 1992. 1991 Mathematics Subject Classification. Primary 20F05, 20F06, 20F32, 20F50. Lectures on results of this note were given at the University of Utah (October 30, 1991, January 9, 16, 23, 30, 1992), the University of Wisconsin-Parkside (February 6, 1992), the City University of New York (February 7, 1992), the University of Nebraska-Lincoln (February 13, 14, 1992), the Kent State University (March 2, 1992), the University of Florida-Gainesville (March 23, 1992). It is the pleasure of the author to thank these universities for sponsoring his visits as well as to express his gratitude to Professors G. Baumslag, B. Chandler, P. Enflo, S. Gagola, S. Gersten, J. Keesling, A. Lichtman, S. Margolis, J. Meakin, G. Robinson, and J. Thompson for their interest in this work. ©1992 American Mathematical Society 0273-0979/92 $1.00 + $.25 per page