The Groups and Nilpotent Lie Rings of Order p⁸ with Maximal Class

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.

Keywords:

• p-group
• maximal class
• Lie p-ring
• Higman’s PORC conjecture

2020 Mathematics Subject Classification:

• 20D15
• 17B30

Acknowledgments

Our classification of the nilpotent Lie rings of order p ⁸ with maximal class ($p\ge 5$) is essentially a hand calculation, with some computer assistance with Magma [3]. To eliminate errors we used Eamonn O’Brien’s p-group generation algorithm [10] in Magma to compute the groups of order p ⁸ with maximal class for $11\le p\le 43$, and confirmed that the number of groups in these cases agreed with the PORC formula given in Theorem 1. We also used Serena Cicalò and Willem de Graaf’s implementation of the Lazard correspondence in their GAP package LieRing [4] to obtain the groups corresponding to the nilpotent Lie rings in our database, so that we could compare them with the groups provided by the p-group generation algorithm. We used Eamonn O’Brien’s Standard Presentation function in Magma to prove that the two sets of groups are identical (up to isomorphism) for $11\le p\le 43$.

Additional information

Funding

The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST), No. 2019R1A6A1A10073437.