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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.

Copyright 1999 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 112-122
• Publisher Identifier: S 1079-6762(99)00069-4
• 1991 Mathematics Subject Classification. Primary 20D15, 11M41; Secondary 03C10, 14E15, 11M45
• Key words and phrases.
• Received by the editors March April 19, 1999
• Posted on August 30, 1999
• Communicated by Efim Zelmanov
• Comments (When Available)

Marcus du Sautoy

DPMMS, 16 Mill Lane, Cambridge CB2 1SB, UK

E-mail address: dusautoy@dpmms.cam.ac.uk