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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 algebras semisimple-algebras rather than groups?

2 added 72 characters in body

The odds of two random elements of a group commuting is the number of conjugacy classes of the group

$$\frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = c(G)$$\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 algebras rather than groups?

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# 5/8 bound in group theory

The odds of two random elements of a group commuting is the number of conjugacy classes of the group

$$\frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = c(G)$$

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 algebras rather than groups?
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