## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.math.psu.edu/era/.
**

Zeta functions and counting finite p-groups
**This journal is archived by the American Mathematical
Society. The master copy is available at
http://www.ams.org/era/
**

## Zeta functions and counting finite p-groups

### Marcus du Sautoy

**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.

*Copyright 1999 American Mathematical Society*

**Retrieve entire article **

#### 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`

*Electronic Research Announcements of the AMS *Home page