14.76 (1999)

Does there exist an absolute constant $c$ such that any finite $p$-group $P$ has an abelian section $A$ satisfying $\lvert A \rvert^c > \lvert P \rvert$?

By a result of A. Yu. Olshanskii (Math. Notes, 23 (1978), 183–185) we cannot require $A$ to be a subgroup. By a result of J. G. Thompson (J. Algebra, 13 (1969), 149–151) the existence of such a section $A$ would imply the existence of a class 2 subgroup $H$ of $P$ with $\lvert H \rvert^c > \lvert P \rvert$.