1. Introduction. The following elementary logical problem was a question in the William Lowell Putnam Mathematical Competition held in March 1953 (1): Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn, and then painted, some segments red, some blue. Prove that some triangle has all its sides the same color.

1. Introduction. After establishing a duplication principle (§2) which enables us to derive a group of order 2h with k + 1 involutory generators from any group of order h with k generators, we shall prove the following combination principle :

Let and be two groups, each having k generators, such that the rth generators of and have relatively prime periods for each r from 1 to k.

1. Introduction. This note shows how a certain infinite graph of degree three, discovered by Laves in connection with crystal structure, can be inscribed (in sixteen ways, all alike) in an infinite regular skew polyhedron which has square faces, six at each vertex. One-eighth of the vertices of the polyhedron are vertices of the graph, and the three edges of the graph that meet at such a vertex are diagonals of alternate squares. Thus either diagonal of any face of the polyhedron can serve as an edge, and the whole graph can then be completed in a unique manner.

In (1), using the theory of group representations, Brauer studied groups of finite order g containing elements A of prime period p which commute only with their own powers Ai. If is a p-Sylow subgroup of , the normalizer can be generated by A and another element B such that

1.1,

where ϒ is a primitive root (mod p), and t and q are positive integers such that

1.2 tq = p − 1.

The concept of an s-ply transitive (1 ≤ s ≤ n) permutation group on n symbols is of considerable importance in the classical theory of finite permutation groups, which was in the height of its development in the period around the turn of the century. The obvious generalization to a permutation group which is s set-transitive (i.e., a group which, for each pair of s-element unordered subsets S, T of the given n symbols, contains a permutation which carries S into T) seems to have received little attention.

1. Introduction. In (1) we have introduced the notions of “construction” and “ generic acyclicity” in order to determine a homology theory for any class of multiplicative systems defined by identities. Among these classes the most interesting one is the class of associative and commutative systems II with a unit element (containing the class of abelian groups).

If m ∈ P where P is a p.i.r. (principal ideal ring), then P/ {m} is a commutative ring with unit element. The elements of this ring are designated by ā where a ∈ P. The set of square matrices of order n with elements in P/ {m} forms a ring with unit element. The units in this ring are the unimodular matrices, i.e., the matrices whose determinants are units of P/ {m}.

Wedderburn, in 1905, proved that there are no finite skew-fields (5). Wedderburn's result has also been proved by Dickson, Artin, Witt, and Zassenhaus (2; 1; 6; 7); however, it seems to the author that the proofs so far given introduce concepts not obviously related to the theorem. It is the purpose of this note to use a result of Cartan, which was later proved in greater generality by Hua (4), to give a simpler and more direct version of the proof of Zassenhaus.

The average order of the Euler function ϕ(n), the number of integers less than n which are relatively prime to n, raises many difficult and still unanswered questions. Thus, for

1.1,

and

1.2,

it is known that R(x) = O(x log x) and H(x) = O(log x). However, though these results are quite old, they were not improved until recently.

1. Introduction. To Schottky is due the theorem which states that a function F(Z), regular and not taking the values 0 and 1 in |Z| < 1 and for which F(0) = a0, is bounded in absolute value in |Z| ≤ r, 0 ≤ r < 1, by a number depending only on a0 and r. Let K(a0 r) denote the best possible bound in this result. Various authors have dealt with the problem of giving an explicit estimate for this bound.

1. Introduction. Let S denote the family of functions f(z) regular and univalent in ∣z∣ < 1, with the expansion f(z) = z + a2z2 + … about z = 0, and let Af denote the area of the intersection of the open circle ∣ω∣ < 1 with Df, the image of ∣z∣ < 1 under f(z). A few years ago one of the authors (1) proved that if

1

then

2.

1. Introduction. Widder has pointed out (2, p. 219) in connection with Wiener's fundamental work on the operational calculus (1, pp. 557-584), that the convolution transform

(1.1)

will be inverted by the operator DE (D), where D = d/dx, and

,

where a suitable interpretation must be found for E(D).

1. Introduction. The literature of the Laplace transformation contains many examples of inversion operators. Particular attention has been given in this literature to the so-called “real” inversion operators, that is, those operators which make use of values of the generating function arising only from real values of the independent variable.

1. Introduction. There are many problems, particularly in circuit theory, where the inverse of a Laplace transform is required but only for large values of the independent variable. For example (2, p. 66) the Laplace transform

of the current function for a semi-infinite cable under certain conditions turns out to be

.

1. Introduction. The potential theory of real harmonic tensors, which was first studied by Hodge (5), offers a variety of problems by no means all of which have yet been examined. In the present paper there are formulated the solutions of some boundary value problems for the Poisson equations associated with coclosed harmonic forms. These problems include as special cases a number of previous results on coclosed harmonic forms and harmonic fields.

1. Introduction. Recently two Japanese authors (1) gave a beautifully simple proof of Selberg's fundamental lemma in the theory of distribution of primes. The proof is based on a curious twist in the Möbius inversion formula. The object of this note is to show that their proof may be extended to a proof of the result for algebraic fields corresponding to Selberg's lemma. Shapiro (2) has already derived this result using Selberg's methods and deduced as a consequence the prime ideal theorem.

1. Introduction. The following remarks concern two different parts of the theory of extreme quadratic forms. In §2, I shall give a new proof for the theorem of Voronoï (4), which asserts that a form is extreme if and only if it is both perfect and eutactic. (For the definitions see, e.g., Coxeter (2) or the text below.) There is indeed a comparatively simple proof in Bachmann's Zahlentheorie, IV, 2. For two reasons, however, it may not be useless to communicate another proof here.

Let ƒ(x1, … , xn) = Σaijxixj be a positive definite quadratic form of determinant D = |aij|, and let M be the minimum of f for integral x1, … , xn not all zero. The form ƒ is said to be extreme if the ratio Mn/D does not increase when the coefficients aij of f suffer any sufficiently small variation.

1. Introduction. If m and n are two integers chosen at random, the probability that they are relatively prime (2, p. 267) is 6π-2. This result may still hold when m and n are functionally related. Thus, Watson (3) recently proved that for α irrational, the positive integers n for which (n, [αn]) = 1, have density 6π-2. A different proof of a slightly more general result was given by Estermann (1).