Automorphisms In Symmetric Balanced Incomplete Block Designs: A Comprehensive Analysis

This study presents a permutation α on a set X using the disjoint cycle representation, with each cycle in the representation on X. This paper considers a symmetric balanced incomplete block design (v, k, λ) as a B-cyclic design with an automorphism that permutes its objects and blocks in a cycle of length v. The results indicate that a cyclic automorphism and a set of objects in a single block completely determine the entire experimental design. The residues modulo v corresponding to the automorphism of the cyclic design, as being the objects and block subscripts respectively have equally been discussed with their properties stated and proved.