Volume 18, Issue 51 (5-2005)
Abstract
Let be a set and let be the set of subsets of . The pair in which is a collection of elements of (blocks) is called a design if every element of appears in , times. Is called a symmetric design if . In a symmetric design, each element of appears times in blocks of . A mapping between two designs and is an isomorphism if is a one-to-one correspondence and . Every isomorphism of a design, , to itself is called an automorphism. The set of all automorphisms of a design with the natural composition rule among mappings forms the automorphism group of the design, and is denoted by . Let be an automorphism of a design , we define , and . In this paper we study the automorphism group of a symmetric design with , and prove the following basic theorem. Theorem. If is a fixed block of a symmetric design, , which have fixed points, then i) ii) There is a symmetric design in the structure of this design. In the particular case we study the automorphism group of a possible symmetric design. The existence or of a symmetric design is unknown. We prove that Theorem. If is a possible symmetric design, then , in which . Also if , and i) If , then ii) If , then iii) If , then .
