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?

If $c(G)> 5|G|/8$, then the average character has a dimension-squared of less than $8/5$, so at least $4/5$ of the characters are dimension $1$ (since the next-smallest dimension-squared is $4$), so the abelianization, which has one element for each 1-dimensional character, is more than half the size of the group, so the commutator subgroup has size smaller than $2$ and so is trivial. 