Difference sets in a finite group
1955; American Mathematical Society; Volume: 78; Issue: 2 Linguagem: Inglês
10.1090/s0002-9947-1955-0069791-3
ISSN1088-6850
Autores Tópico(s)Financial Crisis of the 21st Century
Resumo1. Introduction.Let v, k, X be integers with v>k>X>0.By a (j;, k, X)system II (or, more briefly, a X-plane II) we mean a set II of 2v elements (v of which are called points and v, lines) together with an incidence relation, such that every line contains exactly k distinct points and every two distinct lines contain exactly X common points.Ryser(2) [ll] has proved that (1.1) X(v -1) = k(k -1).Using (1.1), Chowla and Ryser [5] have proved that dually every point lies on exactly k distinct lines and every two distinct points lie on exactly X common lines.Such a system is best known as a symmetric balanced incomplete block design.The present terminology reflects the fact that a 1-plane (X = l) is a finite projective plane.A collineation oi a X-plane II is a one-to-one mapping of the 2v elements of II upon themselves which maps points upon points, lines upon lines and preserves incidence.A X-plane LI will be called transitive^) if there exists a group G of collineations of II such that for each pair P, Q of points of II there is one and only one x of G satisfying Px = Q.In §2 we show that every transitive X-plane may be formed from a difference set (G, D) consisting of a group G of order v and a subset D oik elements with the following properties:(i) If xGG, X9^l, there are exactly X distinct ordered pairs (du d2) of elements of D such that x = di1d2.(ii) If xGG, xj^l, there are exactly X distinct ordered pairs (d%, di) of elements of D such that x = d3di~1.Here the points of D7 are taken as the elements x of G, the lines of II as the subsets Dx of G; and the point x lies on the line Dy if and only if xGDy.It is convenient to note at this point that either of (i), (ii) implies (1.1).As we shall show in §2, (i), (ii) are in fact equivalent properties.A difference set (G, D) will be called abelian or cyclic provided G is abelian or cyclic.Cyclic difference sets have been studied by various authors [2; 3; 4; 5; 6; 7; 8; 9; 10; 14] but the more general subject seems to be new.Hall's notion of a multiplier is newly characterized in §3.The rest of the paper is
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