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
Full-text PDF Free Access
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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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- M.P.F. du Sautoy and F.J. Grunewald, Counting subgroups of finite index in nilpotent groups of class 2. In preparation.
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- J. Denef and L. van den Dries, $p$-adic and real subanalytic sets, Ann. of Math. 128 (1988), 79–138.
- J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no.1, 201–232.
- J. Denef and F. Loeser, Motivic Igusa zeta functions, J. of Algebraic Geom. 7 (1998), no. 3, 505–537.
- M.P.F. du Sautoy, Finitely generated groups, $p$-adic analytic groups and Poincaré series, Annals of Math. 137 (1993), 639–670.
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- M.P.F. du Sautoy, Counting finite $p$-groups and nilpotent groups, preprint (Cambridge).
- M.P.F. du Sautoy and F.J. Grunewald, Analytic properties of zeta functions and subgroup growth, preprint (M.P.I. Bonn).
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- M.P.F. du Sautoy and F.J. Grunewald, Uniformity for 2 generator free nilpotent groups. In preparation.
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- A. Shalev, The structure of finite $p$-groups: effective proof of the coclass conjectures, Invent. Math. 115 (1994), 315–345.
- C.C. Sims, Enumerating $p$-groups, Proc. Lond. Math. Soc. 15 (1965), 151–166.
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Additional Information
Marcus du Sautoy
Affiliation:
DPMMS, 16 Mill Lane, Cambridge CB2 1SB, UK
Email:
dusautoy@dpmms.cam.ac.uk
Received by editor(s):
April 19, 1999
Published electronically:
August 30, 1999
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 1999
American Mathematical Society
