A tournament T is called symmetric if its automorphism groupis transitive on the points and arcs of 1 . The main result of this paper is that if T is a finite symmetric tournament then T is isomorphic to one of the quadratic residue tournaments formed on the points of a finite field GF (p n ) ,
A graph Γ is called symmetric if its automorphism group is transitive on its vertices and edges. Let p be an odd prime, Z(p) the field of integers modulo p, and Z*(p) = (a ∈ Z(p) | a ≠ 0}, the multiplicative subgroup of Z(p). This paper gives a simple proof of the equivalence of two statements:(1) Γ is a symmetric graph with p vertices, each having degree n ≥ 1;(2) the integer n is an even divisor of p − 1 and Γ is isomorphic to the graph whose vertices are the elements of Z(p) and whose edges are the pairs {a, a+h} where a ∈ Z(p) and h ∈ H, the unique subgroup of Z*(p) of order n.In addition, the automorphism group of Γ is determined.
In Recent Years, many discoveries in the history of Islamic mathematics have not been reported outside the specialist literature, even though they raise issues of interest to a larger audience. Thus, our aim in writing this survey is to provide to scholars of Islamic culture an account of the major themes and discoveries of the last decade of research on the history of mathematics in the Islamic world. However, the subject of mathematics comprised much more than what a modern mathematician might think of as belonging to mathematics, so our survey is an overview of what may best be called the “mathematical sciences” in Islam; that is, in addition to such topics as arithmetic, algebra, and geometry we will also be interested in mechanics, optics, and mathematical instruments.
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