Robert Heffernan - University of Connecticut
Des MacHale - University College Cork
Aine Ni She - Cork Institute of Technology

 ‎We consider two commutativity ratios \Pr(G) and f(G) in a finite group G‎ ‎and examine the properties of G when these ratios are `large'‎. ‎We show that‎ ‎if \Pr(G) > \frac{۷}{۲۴}‎, ‎then G is metabelian and we give threshold‎ ‎results in the cases where G is insoluble and G' is nilpotent‎. ‎We also‎ ‎show that if f(G) > \frac{۱}{۲}‎, ‎then f(G) = \frac{n+۱}{۲n}‎, ‎for some‎ ‎natural number n‎.

commutativity ratios, commuting probability, Finite groups