TITLE: 5/8 bound in group theory
QUESTION [73 upvotes]: The odds of two random elements of a group commuting is the number of conjugacy classes of the group
$$  \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2}  = \frac{c(G)}{|G|}$$
If this number exceeds 5/8, the group is Abelian (I forget which groups realize this bound).  
Is there a character-theoretic proof of this fact?  What is a generalization of this result... maybe it's a result about semisimple-algebras rather than groups?

REPLY [20 votes]: One elementary result using character theory, but going in the other direction, which is proved in the paper of R. Guralnick and myself mentioned in my comment above is that if 
$\{\chi_1, \chi_2, \ldots, \chi_c \}$ are the complex irreducible characters of $G$, where $c = c(G)$ is the numberof conjugacy classes of $G,$ then by Cauchy-Schwarz, we have $\sum_{i=1}^{c} \chi_i(1) \leq \sqrt{c}\sqrt{|G|}$, so that $\frac{c(G)}{|G|} \geq \left( \frac{\sum_{i=1}^{c} \chi_i(1)}{|G|} \right)^{2}.$.