A Connection between The Baker-Hausdorff Formula and a Problem of Burnside

Abstract

Let F be the free group of k generators and let V be the subgroup of F which is generated by the pth powers of all the elements of F. Then the quotient group B = F/V has the property that X' = 1 in B, where X stands for an arbitrary element of B. We shall say that B is the group of k generators defined by the relation X' = 1 (cf. B. H. Neumann [1]). W. Burnside [2] was the first to investigate the problem for which values of k and p the group B will be finite. A connection of this problem with the general theory of groups was pointed out by M. Dehn in 1922. For a full account of this and of related questions cf. R. Baer [3]. We shall confine ourselves to the case where p is a fixed prime number >2, the case p = 2 being trivial. We shall also assume k to be equal to 2; this restriction is not essential but it will render the proofs a little shorter. For some purposes (e.g. in the theory of p-groups; cf. [3] and Ph. Hall [4, 5]) it would be sufficient to solve the restricted problem of Burnside: Does there exist a maximum finite group B* among the set of finite groups B' with two generators and with the property that X' = 1 for every element X of B', such that every B' is a quotient-group of B? It is known (cf. [3] and 0. Gruen [6]) that if there exists a group B* then either B is also finite and isomorphic with B* or there exists an infinite group B., such that Boo has a finite number of generators, X' = 1 in Boo and Boo is identical with its commutator subgroup. The group B* exists if and only if the lower central series B1 (=B), B2 , B3 , , B,, , * of B terminates that is if all the Bn are identical for n > m where Bm either is the unit element or a group B,, . Accordingly, we shall try to find relations in B which follow from X' = 1 and which, roughly speaking, involve commutators rather than pth powers. For related results compare also [3, 4, 6, 7, 8].