Block‐Primitive Rank 3 t‐Designs With Sporadic or Alternating Socle

ABSTRACT In this paper, we classify nontrivial ‐ designs with and admitting an automorphism group that acts primitively with rank 3 on the block set, under the assumption that the socle is a sporadic simple group or an alternating group. We show that, apart from the known symmetric examples, the only possibilities are the 2‐, 3‐, and 4‐designs associated with the Mathieu groups and , together with a small number of 2‐designs on points arising from alternating groups.