The fraction of the bijections generating the near-ring of 0-preserving functions

Let (G, +) be a finite (not necessarily abelian) group. Then M-0(G) := {f : G -> G vertical bar f(0) = 0} is a near-ring, i.e., a group which is also closed under composition of functions. In Theorem 4.1 we give lower and tipper bounds for the fraction of the bijections which generate the near-ring M-0(G). From these bounds we conclude the following: If G has few involutions and the order of G is large, then a high fraction of the bijections generate the near-ring M-0(G). Also the converse holds: If a high fraction of the bijections generate M-0(G), then G has few involutions (compared to the order of G).