Abstract

We classify and count the nilpotent Lie rings of order p⁸ with maximal class for $p\ge 5$. This also provides a classification of the groups of order p⁸ with maximal class for $p\ge 11$ via the Lazard correspondence. We also record the number of nilpotent Lie rings/groups of order pⁿ with maximal class for $n\le 7$ from currently known data and discuss its asymptotic behavior as n grows and its potential connection to Higman’s PORC conjecture.

中文翻译：

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最大类的 p8 阶群和幂零李环

摘要

我们对p ⁸阶的幂零李环进行分类和计数，最大类别为$p\ge 5$. 这也提供了p ⁸阶组的分类，其中最大类别为$p\ge 11$通过拉扎德通信。我们还记录了具有最大类别的幂等p ⁿ 阶的幂零李环/组的数量$n\le 7$从目前已知的数据中讨论它随着n增长的渐近行为及其与 Higman 的 PORC 猜想的潜在联系。