The [odds of two random elements of a group commuting](http://www.ime.usp.br/~rbrito/docs/2318778.pdf) 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?



5/8 bound in group theory
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. 


