Is there a McLaughlin geometry?

Abstract

It has long been an open problem whether or not there exists a partial geometry with parameters ( s , t , α ) = ( 4 , 27 , 2 ) . Such a partial geometry, which we call a McLaughlin geometry, would have the McLaughlin graph as point graph. In this note we use tools from computational group theory and computational graph theory to show that a McLaughlin geometry cannot have certain automorphisms, nor can such a geometry satisfy the Axiom of Pasch.