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