# Abelian subgroups of a simple group

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?

• See here – Derek Holt Mar 21 at 22:29