The Burnside problem on periodic groups

The Burnside problem on periodic groups was posed in 1902 and consists of the simple and natural question: "Is a group finite if it is finitely generated and the relation $x^n=1$ is satisfied identically in it?" This is essentially a question of the connection between periodicity and finiteness in group theory. The problem reduces to the question of the finiteness of the free Burnside groups $B(m,n)$, defined by the identity $x^n=1$. A negative solution of the Burnside problem was obtained in a series of joint papers by the present speaker and P. S. Novikov in 1968. We proved then that non-cyclic periodic groups were infinite for all odd periods $n>4381$. To solve the Burnside problem Novikov and I created a new method, later improved by myself and other authors. The result itself was strengthened to all odd $n>664$ by me (in 1975), and by I. G. Lysenok to all $n$ beginning with 8000. This method also enabled a number of other old and difficult problems in group theory to be solved. These results were all obtained by Russian mathematicians. The lecture discussed the history of the problem and surveyed all the results obtained. It is appropriate to mention here that in his monograph on the history of combinatorial group theory (B. Chandler and W. Magnus, Springer-Verlag, New York 1982), one of the creators of combinatorial group theory, the outstanding American algebraist Wilhelm Magnus, wrote: “A comparison of the influence of Burnside's problem on combinatorial group theory with the influence of Fermat's last theorem on the development of algebraic number theory suggests itself very strongly …”.