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Zeta functions and counting finite $p$-groups


Author: Marcus du Sautoy
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 112-122
MSC (1991): Primary 20D15, 11M41; Secondary 03C10, 14E15, 11M45
DOI: https://doi.org/10.1090/S1079-6762-99-00069-4
Published electronically: August 30, 1999
MathSciNet review: 1715428
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Abstract: We announce proofs of a number of theorems concerning finite $p$-groups and nilpotent groups. These include: (1) the number of $p$-groups of class $c$ on $d$ generators of order $p^n$ satisfies a linear recurrence relation in $n$; (2) for fixed $n$ the number of $p$-groups of order $p^n$ as one varies $p$ is given by counting points on certain varieties mod $p$; (3) an asymptotic formula for the number of finite nilpotent groups of order $n$; (4) the periodicity of trees associated to finite $p$-groups of a fixed coclass (Conjecture P of Newman and O'Brien). The second result offers a new approach to Higman's PORC conjecture. The results are established using zeta functions associated to infinite groups and the concept of definable $p$-adic integrals.


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Additional Information

Marcus du Sautoy
Affiliation: DPMMS, 16 Mill Lane, Cambridge CB2 1SB, UK
Email: dusautoy@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S1079-6762-99-00069-4
Received by editor(s): April 19, 1999
Published electronically: August 30, 1999
Communicated by: Efim Zelmanov
Article copyright: © Copyright 1999 American Mathematical Society

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