The Burnside problem and related topics

This paper gives a survey of results related to the famous Burnside problem on periodic groups. A negative solution of this problem was first published in joint papers of P. S. Novikov and the author in 1968. The theory of transformations of words in free periodic groups that was created in these papers and its various modifications give a very productive approach to the investigation of hard problems in group theory. In 1950 the Burnside problem gave rise to another problem on finite periodic groups, formulated by Magnus and called by him the restricted Burnside problem. Here it is called the Burnside–Magnus problem. In the Burnside problem the question of local finiteness of periodic groups of a given exponent was posed, but the Burnside–Magnus problem is the question of the existence of a maximal finite periodic group $R(m,n)$ of a fixed period $n$ with a given number $m$ of generators. These problems complement each other. The publication in a joint paper by the author and Razborov in 1987 of the first effective proof of the well-known result of Kostrikin on the existence of a maximal group $R(m,n)$ for any prime $n$, together with an indication of primitive recursive upper bounds for the orders of these groups, stimulated investigations of the Burnside–Magnus problem as well. Very soon other effective proofs appeared, and then Zel'manov extended Kostrikin's result to the case when $n$ is any power of a prime number. These results are discussed in the last section of this paper.
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