jats:titleAbstract</jats:title>jats:pLet jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972722000211_inline2.png" />jats:tex-math $\eta (G)$ </jats:tex-math></jats:alternatives></jats:inline-formula> be the number of conjugacy classes of maximal cyclic subgroups of jats:italicG</jats:italic>. We prove that if jats:italicG</jats:italic> is a jats:italicp</jats:italic>-group of order jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972722000211_inline3.png" />jats:tex-math $pn$ </jats:tex-math></jats:alternatives></jats:inline-formula> and nilpotence class jats:italicl</jats:italic>, then jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972722000211_inline4.png" />jats:tex-math $\eta (G)$ </jats:tex-math></jats:alternatives></jats:inline-formula> is bounded below by a linear function in jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972722000211_inline5.png" />jats:tex-math $n/l$ </jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>