CONJUGACY CLASSES of MAXIMAL CYCLIC SUBGROUPS and NILPOTENCE CLASS of p-GROUPS
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Publication Date
2022-03-21Journal Title
Bulletin of the Australian Mathematical Society
ISSN
0004-9727
Publisher
Cambridge University Press (CUP)
Pages
1-7
Type
Article
This Version
AM
Metadata
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Bianchi, M., Camina, R., & Lewis, M. (2022). CONJUGACY CLASSES of MAXIMAL CYCLIC SUBGROUPS and NILPOTENCE CLASS of p-GROUPS. Bulletin of the Australian Mathematical Society, 1-7. https://doi.org/10.1017/S0004972722000211
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>
Keywords
group covering, nilpotence class, group exponent
Embargo Lift Date
2022-09-21
Identifiers
External DOI: https://doi.org/10.1017/S0004972722000211
This record's URL: https://www.repository.cam.ac.uk/handle/1810/336044
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International
Licence URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
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