From: Andrea Caranti
Subject: Re: Two Totally Different Kinds of Questions
Date: Sun, 21 Mar 1999 20:22:06 +0100
Newsgroups: sci.math
Keywords: How many groups of order 2^n (n through 8)
David Harden wrote:
> 2. Let G(n) denote the number of groups of order n up to isomorphism.
> Where are the most extensive calculations of G(n) that have been made?
> Do we know G(128)? G(256)? G(512)? G(1024)? Are there programs for doing
> this? How fast are they? How much thought has been given to this? Who
> wrote "Groups of order 2^n, n<=6"?
Groups of order 256 have been classified by Eamonn O'Brien using an
algorithm/programme called "p-group enumeration". There are 56092 of them. I
can give you a precise reference tomorrow if you wish. The programme (which
is part of the so-called "ANU p-quotient program", a.k.a. pq) is available
from the math Web site of the Australian National University at Canberra:
http://wwwmaths.anu.edu.au/services/ftp.html
together with a GAP library of groups of order dividing 256.
There are 10 494 213 groups of order 512: they have been recently counted by
Bettina Eick and Eamonn O'Brien. O'Brien is the right person to contact for
further information on such enumeration questions. He's at the University of
Auckland, NZ.
Andreas Caranti
==============================================================================
From: "A. Caranti"
Subject: Re: Enumeration/Construction of Finite P-Groups?
Date: Tue, 01 Jun 1999 17:52:11 +0200
Newsgroups: sci.math.research
Jim Heckman wrote:
> (2) Is there an easy way to construct all of the p-groups of a given order
> p^n?
In principle, yes. The p-group generation algorithm, by E. A. O'Brien
does just that. This is implemented as part of the Australian National
University p-Quotient Program, by G. Havas, M.F. Newman and E. A.
O'Brien.
O'Brien used this to enumerate groups of order 2^8. There are 56 092 of
them. recently, B. Eick and O'Brien have COUNTED the number of groups of
order 2^9. There are 10 494 213 such groups. It is clear that there are
practical limits to what you can do here.
Andreas
References (in mild TeX):
ANU p-Quotient program available under
ftp://ftpmaths.anu.edu.au/pub/algebra
E. A. O'Brien, The $p$-group generation algorithm, J. Symbolic Comput.
{\bf 9} (1990), no.~5-6, 677--698; MR 91j:20050
E. A. O'Brien, The groups of order $256$, J. Algebra {\bf 143} (1991),
no.~1, 219--235; MR 93e:20029
B. Eick and E. A. O'Brien, The groups of order $512$, in {\it
Algorithmic algebra and number theory (Heidelberg, 1997)}, 379--380,
Springer, Berlin,
; CNO CMP 1 672 078
B. Eick and E. A. Brien, Enumerating $p$-groups, to appear, 1999
==============================================================================
From: sfinch@mathsoft.com (Steven Finch)
Subject: Re: Enumeration/Construction of Finite P-Groups?
Date: Tue, 01 Jun 1999 09:59:51 -0400
Newsgroups: sci.math.research
>Is the number of finite p-groups (i.e., those of order a
>power of a prime) known for all orders q=p^n?
Let G(k) denote the number of non-isomorphic groups
of order k. Higman (1960) and Sims (1965) proved
that
log(G(p^n)) = n^3 * (2/27 + O(n^(-1/3)))
as n approaches infinity and p is fixed. See the
survey "Counting finite groups" by M. Ram Murty,
available online at
http://www.math.mcgill.ca/~murty/
as well as
L. Pyber, Group enumeration and where
it leads us, from European Congress of
Mathematics, vol. 2, Budapest 1996,
Birkhäuser Verlag, 1998; pp. 187-199.
It's unlikely that an explicit formula for G(p^n)
will ever be found.
Steve Finch
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