Let $G$ be a finite non-abelian simple group, and $A \leqslant G$ be an abelian subgroup. How large $A$ can be? There exists any bound of the type $|A| \leq |G|^r$ for some $r<1$? How can I prove it?

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    $\begingroup$ See here $\endgroup$ – Derek Holt Mar 21 at 22:29

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