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 ********************************************************************* Steven Finch sfinch@mathsoft.com MathSoft, Inc. Favorite Mathematical Constants 101 Main St. Unsolved Mathematics Problems Cambridge, MA 02142 MathSoft Math Puzzle Page USA http://www.mathsoft.com/asolve/sfinch.html