Restrictions on commutativity ratios in finite groups

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 $n$.

Keywords

• commutativity ratios
• commuting probability
• Finite groups