CONJUGACY CLASSES of MAXIMAL CYCLIC SUBGROUPS and NILPOTENCE CLASS of p-GROUPS

Abstract

<jats:title>Abstract</jats:title> <jats:p>Let <jats:inline-formula> <jats: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:italic>G</jats:italic>. We prove that if <jats:italic>G</jats:italic> is a <jats:italic>p</jats:italic>-group of order <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972722000211_inline3.png" /> <jats:tex-math> $p^n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and nilpotence class <jats:italic>l</jats:italic>, then <jats:inline-formula> <jats: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-formula> <jats: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>