# Restrictions on commutativity ratios in finite groups

Document Type: Research Paper

Authors

1 University of Connecticut

2 University College Cork

3 Cork Institute of Technology

Abstract

‎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 $n$‎.

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