We consider $G$, a noncommutative group which contains $n$ elements.
Show the probability that $2$ elements of $G$ commute is lower than $\frac{5}{8}$.
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2Seems to be Theorem 1.7 of this note that showed up by a search. – fuglede Aug 08 '14 at 13:50
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Paragraph 2 of that paper states exactly the result you want to prove, and claims that the paper contains a proof. – MJD Aug 08 '14 at 14:25