  Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so
that $|G:Z(G)| = p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let
$$p^l = {\rm max} \{|Z(C_G (g)):Z(G)| : g \in G \setminus Z(G)\},$$ $$p^b =
{\rm max} \{|cl(g)| : g \in G \setminus Z(G) \},$$ and $p^a = |A:Z(G)|$. Then
we show that $a \ge n/(b+l)$.
