The Burnside Problem on periodic groups for odd exponents \(n>100\)

S. Adian

In 1902, W.Burnside formulated the following problem:

Is every group generated by a finite number of generators and satisfying the identical relation \(x^n = 1\) finite?

Maximal periodic groups \(B(r,n)\) with \(r\) generators satisfying the identical relation \(x^n = 1\) are called the free Burnside groups of exponent \(n\).

During several decades many mathematicians from different countries studied this problem.

In 1950, W. Magnus formulated a special question on the existence of a maximal finite quotient group of the group \(B(r,n)\) for a given pair \((r,n)\). Magnus named this question "the restricted Burnside problem".

A negative solution of the full (nonrestricted) Burnside problem was given in 1968 by P. S. Novikov and S. I. Adian. It was proved that the groups \(B(r,n)\) are infinite for any \(r > 1\) and odd \(n \ge 4381\).

In 1975, the author published a book where he presented an improved and generalized version of the Novikov–Adian theory for odd exponents \(n \ge 665\) and established some other applications of the method.

In this talk, we introduce a new simplified modification of the Novikov–Adian theory that allows us to give a shorter proof and stronger results for odd exponents. The main result is the following new theorem.

Theorem. The free Burnside groups \(B(m,n)\) are infinite for any odd exponent \(n > 100\).

A detailed survey of investigations on the Burnside problem and on the restricted Burnside problem can be found in the survey paper [1].

References

1. S. I. Adian, The Burnside problem and related questions, Russian Math. Surveys, 65 N5, (2010), 805–855.

See also the author's pdf version: