If $G$ is a finite simple group then is it true that an abelian subgroup $H$ of $G$ of maximal order has order $|H| < |G|^{\frac{1}{3}}$? If so, could you please point me to a reference for this, or where a proof is given? Secondly, if $G$ is now a finite nilpotent group of class $2$ and $H$ is again an abelian subgroup of $G$ of maximal order, then is it true that $|H| \leq \sqrt{|G|}$? If so, is there a reference or proof for this?


Thanks in advance for any help.


Sandeep Murthy



How large can abelian subgroups of class 2 nilpotent groups or simple groups be?
The answer to both questions is *no*:


Counterexample to first assertion: $G = {\rm A}\_5$, $H = \langle (1,2,3,4,5) \rangle$.


Counterexample to second assertion: $G = \langle (1,2,3,4), (1,3), (5,6) \rangle$, $H = \langle (1,2,3,4), (5,6) \rangle$.


