Our title has become something of a misnomer, however we retain it since drafts of this note have been quoted with it.

A consequence of Schreier's formula is that if G is a subgroup of the free group F of rank n < 1 and rank G ≦ n, then G = F or G is of infinite index in F. However, if S is a free sovlvable group of derived length I < 1 and H is a subgroup of S which is free solvable of the same length, then the rank of H does not exceed the rank of S. These observations led G. Baumslag to conjecture that if H is of finite index in S then H = S. In fact, we have sharper results in two directions. If H and S are free solvable of the same length, not only is H of infinite index in S, but δ1−1(S)/δ1−1(H) is torsion-free. In another direction we need not assume that S is free solvable, only that s is torsion-free and of derived length l (l > 1) and that H is not cyclic. Thus Stallings' theorem [11] that a finitely generated torsionfree group with a free subgroup of finite index is itself free has an even stronger counterpart in the variety of groups solvable of length at most l (l > 1): a torsionfree group in that variety with a non-cyclic free subgroup of finite index coincides with this subgroup. The proof relies on the following theorem: If S is a free solvable group, J is the group of automorphisms of S which induce the identity on S/S', and I is the group of inner automorphisms of S, then J/I is torsion-free. The proofs of these theorems form the bulk of the first four sections.

This paper presents an example to show that the lattice of subvarieties of 3, 9 Λ11 is non-distributive. The example is used further to show that a certain ‘canonic” description for non-nilpotent subvarieties of pp2, p prime, is generally not unique.

Let T denote the circle group, C the set of continuous complex-valued functions on T, and A the set of f ∈ C having absolutely convergent Fourier series:

I standing for the set of integers.

In [1] D. W. Miller and the author established necessary and sufficient conditions for the existence of a cancellative (ideal) extension of a commutative cancellative semigroup by a cyclic group with zero. The purpose of this paper is to extend these results to cancellative extensions by any finitely generated Abelian group with zero and to establish in this general case conditions under which two such extensions are equivalent.

A. H. Clifford [1], [2] has shown that all finite dimensional representations of a completely 0-simple semigroup S over a field Φ.phi; can be obtained as extensions of those of its maximal subgroups and has given a method for constructing all such representations. This representation theory depends strongly on the fact the representations under consideration are finite dimensional and is not adequate to deal with the infinite dimensional case or with representations over arbitrary rings. In order to determine the structure of the (contracted) algebra Φ(S) of S modulo its radical, one has to consider representations which are not finite dimensional or over fields; c.f. ‘6’. Hence Clifford's theory does not suffice for this purpose.

It is well known (e.g. Stone [1]) that the Stone-Weierstrass approximation theorem can be used to prove the completeness of various systems of orthogonal polynomials, e.g. Chebyshev polynomials. In this paper, Stone's theorem is used to prove a more general completeness theorem, which includes as special cases Plancherel's theorem, the corresponding theorem for Hankel transforms, the completeness of various polynomial systems, and certain expansions in Jacobian elliptic functions. The essential feature common to all these systems is a certain algebraic structure — if S is an appropriate vector space spanned by orthogonal functions, then the algebra A generated by S is contained in the closure of S in a suitable norm.

In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimum m+(f) of the non-negative values1 of an indefinite quadratic form f in n variables, of given determinant det(f) ≠ 0 and of signature s. In particular it was announced (and later proved — see [2]) that m+(f) ≦ (16/5)+ for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is that m+(f) ≦ (256/135)+ for quaternary quadratic forms of determinant — 1 and signature — 2.

Let n1 < n2 < … be the sequence of positive integers whose base b representations involve the digit t ≦ b —1 at most d—1 times. B. D. Craven [2] shows that Σi1/ni converges by giving an upper bound for this sum as a function of b and d.

We say a matrix H of order 4t is Hadamard if each entry is 1 or — 1, and the inner product of any two rows is zero. We shall consider only Hadamard matrices in normal form with the first row consisting solely of 1 while any two of the remaining rows have the property that hik = hjk = 1, hik = –hjk = 1, —hik = hjk = 1, and –hik = –hjk = 1 each occur t times. We can induce a further normalization by choosing the second row of H to have the first 2t entries 1 and the last 2t entries — 1, and this will be the standard form we consider. We now call the submatrix obtained by deleting the first two rows of H and the first It columns. H therefore is of dimension 4t — 2 x 2t. We prove the following theorem:

Most known homology theories (e.g. the homology of modules, rings, groups, sheaves, …) have been found to be special cases of a general theory proposed by M. Barr and J. Beck [1], [2]. The aim of this paper is to show that the cohomology of a local group, as defined by W. T. van Est [4], also fits the scheme of Barr and Beck. At the same time it will be shown that local group cohomology is a relative derived functor in the sense of S. Eilenberg and J. C. Moore [3].