Conjugacy classes of maximal cyclic subgroups and nilpotence class of $p$-groupsIn this paper, we set $η(G)$ to be the number of conjugacy classes of
maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order
$p^n$ and nilpotence class $l$, then $η(G)$ is bounded below by a linear
function in $n/l$.
arxiv.org