3‐Designs From PSL(2,q) With Cyclic Starter Blocks

ABSTRACT We consider when the projective special linear group over a finite field defines a block‐transitive 3‐design with a starter block which is a multiplicative subgroup of the field. For a prime power , we will show that the multiplicative subgroup of order 5 is a starter block of a 3‐design if and only if the multiplicative subgroup of order 10 is a starter block of a 3‐design. The former is the family of 3‐ designs investigated by Li, Deng and Zhang, while the latter appear in a different context by Bonnecaze and Solé for the case . We also show a similar equivalence for multiplicative subgroups of orders 13 and 26 for a prime power .