University of Isfahan International Journal of Group Theory 2251-7650 3 4 2014 12 01 Restrictions on commutativity ratios in finite groups 1 12 4570 10.22108/ijgt.2014.4570 EN Robert Heffernan University of Connecticut Des MacHale University College Cork Aine Ni She Cork Institute of Technology Journal Article 2013 09 05  ‎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 $\Pr(G) > \frac{7}{24}$‎, ‎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 $f(G) > \frac{1}{2}$‎, ‎then $f(G) = \frac{n+1}{2n}$‎, ‎for some‎ ‎natural number $n$‎. https://ijgt.ui.ac.ir/article_4570_55d0f1553f00e86de09233d4129f5a8f.pdf