ON THE RESTRICTED BURNSIDE PROBLEM

Abstract

After many unsuccessful attempts to obtain a proof in the late 30s-early 40s the following weaker version of The Burnside Problem was studied: Is it true that there are only finitely many 7??-generated finite groups of exponent nl In other words the question is whether there exists a universal finite m-generated group of exponent n having all other finite m-generated groups of exponent n as homomorphic images. Later (thanks to W. Magnus [35]) this question became known as The Restricted Burnside Problem. In 1964 E. S. Golod gave a negative answer to The General Burnside Problem (cf. [9]). Since then a considerable array of infinitely generated periodic groups was constructed by other authors (cf. Alyoshin [2], Suschansky [44], Grigorchuk [ll],Gupta-Sidki [54]). In 1968 P. S. Novikov and S. I. Adian [39] constructed counter-examples to The Burnside Problem for groups of odd exponents n > 4381 (now for odd exponents n > 115, cf. I. Lysenok [33]). Olshansky's Monsters (cf. [40]) shows how wildly periodic groups may behave. At the same time there were two major reasons to believe that The Restricted Burnside Problem would have a positive solution. One of these reasons was the reduction theorem obtained by Ph. Hall and G. Higman [14]. Let n = . ..pf, where /;,• are distinct prime numbers, iq > 1, and assume that (a) The Restricted Burnside Problem for groups of exponents pf has a positive solution, (b) there are only a finite number of finite simple groups of exponent n, (c) the factor group Out(G) = Aut(G)/Inn(G) is solvable for any finite simple group of exponent n. Then The Restricted Burnside Problem for groups of exponent n also has positive solution. Another reason was the close relation of The Problem to Lie algebras. Suppose n = p, where p is a prime number. Then the finite group G of exponent p is clearly nilpotent. It is easy to see that it is sufficient to find an upper bound