A lower bound on the size of maximal abelian subgroups

Abstract

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 $$p^l=\max \{|Z(C_G(g)):Z(G)|:g\in G\setminus Z(G)\},$$ $$p^b=\max \{|cl(g)|:g\in G\setminus Z(G)\},$$ and $p^a=|A:Z(G)|$. Then we show that $a\ge n/(b+l)$.