Henry B. Mann

On linear relations between roots of unity

In a recent paper I. J. Schoenberg [1] considered relations where the a v are rational integers and the ζ v are roots of unity. We may in (1) replace all negative coefficients a v by − a v replacing at the same time ζ v by −ζ v so that we may, if it is convenient, assume that all a v are positive. If we do this and arrange the ζ r so that their arguments do not decrease with v then (1) can, as suggested by Schoenberg (oral communication) be interpreted as a convex polygon with integral sides whose angles are rational when measured in degrees. Accordingly we shall call a relation (1) a polygon if all a v are non-negative. We shall call a polygon (1) k -sided if all a v are positive. The polygon is called degenerate if two of the ζ v are equal. Schoenberg calls these polygons polar rational polygons (abbreviated prp) because the vectors composing them have rational coordinates in their polar representations. Schoenberg showed that every prp can be obtained as a linear combination with integral positive or negative coefficients of regular p -gons where p is a prime.

On the p-rank of the design matrix of a difference set

Abstract : The rank mod p (a prime) of the incidence matrix of the Euclidean and projective hyperplanes is determined. Similar results are obtained for certain other design matrices. (Author)

On orthogonal Latin squares

An m-sided Latin square is an arrangement of the numbers 1, 2, • • • , m into m rows and m columns in such a way that no row and no column contains any number twice. Two Latin squares are said to be orthogonal if when one is superimposed upon the other every ordered pair of numbers occurs once in the resulting square. Various methods have been devised for the construction of sets of orthogonal squares. However no method has as yet been given which would yield all possible sets of orthogonal Latin squares. In constructing orthogonal sets it is of value to have simple criteria which enable us to decide whether a given Latin square can be a member of an orthogonal pair. A Latin square to which an orthogonal square exists will be called a basis square. In this note we shall derive two simple necessary conditions for a square to be a basis square.

Two addition theorems

Abstract The following theorems are proved: o (1) Let A⊕B=A∪B∪(A+B). If G is a finite Abelian group and A 1 +…+A k subsets of G with |A 1 |+…+|A k |≥|G| then either A 1 ⊕…⊕A k =G or 0∈A 2 +…+A k . For k=2 this statement is true for any group. (3) Let a 1 , …, a p+k−1 be a sequence of p+k−1 integers. Then it is possible to select k distinct indices i 1 , …, i k such that a i 1 +…+a i k ≡0(mod p ). By means of (2), the proof of a theorem of Erdos, Ginzburg, and Ziv can be considerably simplified.

On the number of information symbols in Bose-Chaudhuri codes*

Let α be a primitive root of G. F. ( q m ). Let I(m, v) be the number of information symbols of the code with parity check matrix ( α \mij ), i = 1, …, v, j = 0, …, q m − 2. Let v = q λ , m − λ = r . Then for sufficiently large m we have {fx0153-1} where 〈 c 〉 denotes the nearest integer to c and ρ is the positive root of the equation x r = ( q − 1) ( x r−1 + … + 1). For small values of m we have I(m, v) = ρ m + e where | e \t| ≦ ( r − 1) τ m , ″> τ 〈 m ≧ r − 1, are also given. The first contains r terms, the second [ m/r + 1] terms, where [ c ] denotes the largest integer not exceeding c .

Sums of sets in the elementary abelian group of type (p, p)*

Abstract Let S=(α1, …, α2p−1) be a sequence of 2p−1 elements of an Abelian group G of type (p, p). The following theorems are proved: (1) If α1, …, α2p−1 are distinct and not zero, then every element of G is a sum over a subsequence of S and if p>2 then 0 is the sum of a subsequence of α1, …, α2p−2. (2) Let K be a proper subgroup of G. If 0≠αi ∈ K for 1≤i≤p−1, αi∋K for p≤i≤2p−2 then every element of G occurs as a sum over some subsequence of S.

Combinatorial Problems in Finite Abelian Groups

Publisher Summary This chapter presents combinatorial problems in finite Abelian groups. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order. Abelian groups generalize the arithmetic of addition of integers. The concept of an Abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements. The theory of abelian groups is generally simpler than that of their non-Abelian counterparts, and finite Abelian groups are also well understood. Every finite abelian group G can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special case of the fundamental theorem of finitely generated Abelian groups when G has zero rank.

Antisymmetric difference sets

Abstract A difference set D in a group G is called antisymmetric if D ⌣ (−D) = π and D ⌣ (−D) ⌣ (0) = G . If G is Abelian and has an antisymmetric difference set then G is of prime power order, and if G is not elementary Abelian then G has at least two invariant factors whose order is the exponent of G.

Additive group theory—A progress report

is solvable for every r. One easily obtains this result by setting A = B = {x:x = a(p)}. We then have \A\ = \B\ = (p + l)/2 and (2) follows from (1). Applying the C.-D. theorem to the representation of residues by sums of fcth powers one may without loss of generality restrict k to divisors of (p — 1). The C.-D. theorem then gives the result that every residue is a sum of not more than k kth powers. A considerable improvement is possible if one excludes the value k = (p — l)/2. G. A. Vosper [30], [31], [21] refined the C.-D. theorem by completely characterizing those pairs A, B for which

LINEAR EQUATIONS OVER A COMMUTATIVE RING.

Abstract : Let A denote a n x m matrix over a commutative ring R and let x and l be n-rowed column vectors with elements in a module over R. Necessary and sufficient conditions are found that the equation Ax = l has a solution. (Author)