A Remark on Critical Groups

Problem 24 of Hanna Neumann's book [3] reads: Does there exist, for a given integer n > 0, a Cross variety that is generated by its ^-generator groups and contains (&+w)-generator critical groups? In such a variety, is every critical group that needs more than k generators a factor of a kgenerator critical group, or at least of the free group of rank A? I n a recent paper [1], R. G. Burns pointed out that the answer to the first question is an easy affirmative, and asked instead the question which presumably was intended: Given two positive integers k, I, does there exist a variety 23 generated by ^-generator groups and also by a set S of critical groups such that S contains a group G minimally generated by k-\-l elements and S\{G} does not generate S3? The purpose of this note is to record a simple example which shows that the answer to the question of Burns is affirmative at least for k = 2, / = 1, and also that the answer to the second question of Hanna Neumann's Problem 24 is negative. Let 23 be the variety defined by the law x[x, y, u][x, y, u, v]. It can be read off from Bjarni Jonsson's description [2] of the lattice of nilpotent varieties of class at most 3 that 23 has precisely two maximal subvarieties: the subvariety 11 defined by the additional law [x, y, y], and the sub variety SB defined by the additional law [x, y]; moreover, U is certainly not of class 2. Since 9B is defined (within 23) by a two-variable law, it cannot contain the 23-free group F of rank 2; nor can F be contained in 11, for the twogenerator groups of U are all of class at most 2 while 23 contains the wreath product of two cyclic groups of order 3, a two-generator group of class 3. Thus F generates 23. A direct calculation shows that the proper subgroups of F are all of class at most 2: this, and a similar calculation below, is somewhat simplified by the observation that the Frattini subgroup of any group in 23 is contained in the second term of the upper central series of the group. Next, consider the ll-free group H of rank 3, on the free generators a, b, c. As the only relators of these generators are the laws of 11, neither [a, b] nor [a, b, c] can be 1. On the other hand, as 11 has class 3 and [ \.> y]> ] is a law even in 23, the element [a, b] [a, b, c] is central in H. Let N be maximal among the normal subgroups of H which contain [a, b] [a, b, c] but not [a, b, c], and put H/N = G. By construction, G is