Large subgroups of small class in finite p-groups

Suppose p is a prime, P is a finite p-group, and A is an abelian subgroup of P. Does P possess a normal abelian subgroup of the same order as A? J. Alperin showed that the answer is negative in general, but affirmative under certain conditions. In [J. Alperin, G. Glauberman, J. Algebra 203 (1998) 533–566], using an idea from algebraic group theory, he and the author generalized this work under the restriction that A be elementary abelian, and obtained some related results for p-groups and nilpotent Lie algebras over fields of characteristic p. In this paper, we remove the restriction by extending the application of algebraic group theory. We also obtain generalizations to subgroups of small nilpotence class, not necessarily abelian, and to nilpotent Lie algebras over local rings.