Group – Theoretic Analysis in Balanced Incomplete Block Design Structures

The construction of block designs remains a challenging and open problem in combinatorial design theory, with balanced incomplete block designs (BIBDs) representing one of the most extensively studied classes. This research objective is to investigate the existence and structure of BIBDs from an algebraic perspective by examining their compliance with the axioms of finite groups. Both symmetric and non-symmetric BIBDs are considered, and relevant theorems are formulated and proved within the framework of finite group theory. The algebraic structures examined in this study fail to satisfy group axioms under the specified multiplicative operation. However, under addition, they satisfy the axioms of an abelian group, thereby establishing their algebraic compatibility with finite group structures.