From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Finite groups
Date: 14 Feb 1995 05:50:10 GMT
In article ,
P-A wrote:
>Is it possible to determine the number of groups of a given finite order? The
>groups need not to be abelian.
Oh, sure. Let's see, if you want the order to be N then you need only
write down all (N!)^N square tables each of whose rows is a permutation
of {1,...,N}, then toss out those which violate the axioms for the
binary operation of a group. Of course, I assume you only want the groups
up to isomorphism, but it's just a finite task to see if two such tables
describe a finite group.
But of course you were hoping for something more efficient. The answer is
still yes, but just how efficient you can be depends, not on the order N
so much as on its prime factorization. Up to isomorphism, there are (is)
1 group of order p (prime)
2 groups of order p^2 (p prime)
1 group of order pq (p < q distinct primes, gcd(p,q-1)=1)
2 groups of order pq (p < q distinct primes, gcd(p,q-1)=p)
"and so on". The general pattern is that if N = Prod(pi^(ei)) is the
prime factorization of N, then there are few groups of order N if the
ei are small, and really few if there just a few pi too. (The actual
count depends not only on the collection of exponents ei; the number
of groups of order 2^e is not quite the same as the number of groups
of order 3^e, for example).
The bad news is that "few" is a relative term. The number of
groups of order 2^n is (if memory serves)
1, 1, 2, 5, 14, 51, 267, ...
for n=0,1,2,3,4,5,6... In fact I seem to recall a theorem along these
lines: let a_n be the number of non-isomorphic groups of order p^n.
Then log_p (a_n) is asymptotic to (2/27)n^3. [Here "asymptotic" probably
means the ratio of the two sides tends to 1. Sorry, no citation handy.]
I'll add that for solvable groups an enumeration can be made with a
concerted effort using an induction on Sum(ei), although this assumes
that a practical computation of cohomology groups can be carried out,
which is not immediately clear except in principle.
"In theory there's no difference between theory and practice,
but in practice there is."
dave
==============================================================================
Newsgroups: sci.math
Subject: Re: Finite groups
From: mann@vms.huji.ac.il (Avinoam Mann)
Date: 19 Feb 1995 16:54 IDT
In article , pas@mastercs.hv.se (P-A)
writes...
>Is it possible to determine the number of groups of a given finite order? The
>groups need not to be abelian.
>
>P-A
It depends on what you mean by "determine". In principle, you can write
down all possible multiplication tables of n elements, then check which ones
of them define groups, then check these for isomorphism. This shows that the
function f(n), giving the number of (isomorphism types) of groups of order n,
is recursive, or computable. If you want a closed formula, in terms of
elementary functions of n, such a formula probably does not exist. It's more
reasonable to ask if there exists a formula which has as arguments not only n,
but its prime divisors and their multiplicities. I still believe that there is
no such formula in terms of elementary functions, but this would be harder to
establish. For the best asymptotic results, look up a paper of L.Pyber in Ann.
Math. 137(1993). f(n) is at most n^(c(logn)^2), for some universal constant c.
This is best possible, because it is known that if n = p^e is a power of the
prime p, then f(n) = p^((2/27)e^3 + O(e^(8/3))) (I was told that there is a
proof, which I've not seen yet, that the 8/3 can be improved to 5/2). On the
other hand f(n) = 1 if n is prime, and also for infinitely other values of n.
Avinoam Mann
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Number of discrete groups
Date: 17 Feb 1996 04:33:03 GMT
In article <4g2ahb$qal@rzsun02.rrz.uni-hamburg.de>,
Hauke Reddmann wrote:
>I read one post concerning the number of groups with
>element number n,n<200. Call it m(n). Can you give
>an asymptotic formula for m(n) if n => infinity?
It's not that nice a function. If N= 2^(11213) then m(N) is, oh, about
2^(10^11), whereas m(N-1)=1.
[Stale URL deteled -- djr]
dave
==============================================================================
From: qscgz@aol.com (QSCGZ)
Newsgroups: sci.math
Subject: Re: groups of order 48
Date: 30 Nov 1998 14:17:19 GMT
hwatheod@leland.Stanford.EDU (theodore hwa) wrote:
>ortiz (ortiz@ups-albi.fr) wrote:
>: In order to classify all groups G of order 48,
>: I wonder if G is a semi-direct product.
>
>: Now, how to continue ? And how many groups have order 48 ?
>: Where can I find (on the web) tables of small order groups (up to 100)?
>
>http://math.stanford.edu/~goldfarb/math155_html/number_of_groups
>
>has a table of the number of groups of all orders <=200, except for 192.
>
>There are 52 groups of order 48 according to this table.
>
>If you want to know exactly what the groups are, I don't see a way around
>having to find the groups of smaller orders first (16,, 24).
I recently found an extensive library of finite groups in "GAP" .
All these groups are available in compressed form in an ~3MB file.
But presumably you can't easily access them without downloading the
whole GAP-package , which is ~20MB !
Here are the numbers of (nonisomorphic) groups of orders < 1000 :
(except 512 and 768)
0 ---- 0 1 1 1 2 1 2 1 5 2 2 1 5 1 2 1
16 --- 14 1 5 1 5 2 2 1 15 2 2 5 4 1 4 1
32 --- 51 1 2 1 14 1 2 2 14 1 6 1 4 2 2 1
48 --- 52 2 5 1 5 1 15 2 13 2 2 1 13 1 2 4
64 --- 267 1 4 1 5 1 4 1 50 1 2 3 4 1 6 1
80 --- 52 15 2 1 15 1 2 1 12 1 10 1 4 2 2 1
96 --- 231 1 5 2 16 1 4 1 14 2 2 1 45 1 6 2
112 --- 43 1 6 1 5 4 2 1 47 2 2 1 4 5 16 1
128 --- 2328 2 4 1 10 1 2 5 15 1 4 1 11 1 2 1
144 --- 197 1 2 6 5 1 13 1 12 2 4 2 18 1 2 1
160 --- 238 1 55 1 5 2 2 1 57 2 4 5 4 1 4 2
176 --- 42 1 2 1 37 1 4 2 12 1 6 1 4 13 4 1
192 --- 1543 1 2 2 12 1 10 1 52 2 2 2 12 2 2 2
208 --- 51 1 12 1 5 1 2 1 177 1 2 2 15 1 6 1
224 --- 197 6 2 1 15 1 4 2 14 1 16 1 4 2 4 1
240 --- 208 1 5 67 5 2 4 1 12 1 15 1 46 2 2 1
256 --- 56092 1 6 1 15 2 2 1 39 1 4 1 4 1 30 1
272 --- 54 5 2 4 10 1 2 4 40 1 4 1 4 2 4 1
288 --- 1045 2 4 2 5 1 23 1 14 5 2 1 49 2 2 1
304 --- 42 2 10 1 9 2 6 1 61 1 2 4 4 1 4 1
320 --- 1640 1 4 1 176 2 2 2 15 1 12 1 4 5 2 1
336 --- 228 1 5 1 15 1 18 5 12 1 2 1 12 1 10 14
352 --- 195 1 4 2 5 2 2 1 162 2 2 3 11 1 6 1
368 --- 42 2 4 1 15 1 4 7 12 1 60 1 11 2 2 1
384 --- 20169 2 2 4 5 1 12 1 44 1 2 1 30 1 2 5
400 --- 221 1 6 1 5 16 6 1 46 1 6 1 4 1 10 1
416 --- 235 2 4 1 41 1 2 2 14 2 4 1 4 2 4 1
432 --- 775 1 4 1 5 1 6 1 51 13 4 1 18 1 2 1
448 --- 1396 1 34 1 5 2 2 1 54 1 2 5 11 1 12 1
464 --- 51 4 2 1 55 1 4 2 12 1 6 2 11 2 2 1
480 --- 1213 1 2 2 12 1 261 1 14 2 10 1 12 1 4 4
496 --- 42 2 4 1 56 1 2 1 202 2 6 6 4 1 8 1
512 --- ? 15 2 1 15 1 4 1 49 1 10 1 4 6 2 1
528 --- 170 2 4 2 9 1 4 1 12 1 2 2 119 1 2 2
544 --- 246 1 24 1 5 4 16 1 39 1 2 2 4 1 16 1
560 --- 180 1 2 1 10 1 2 49 12 1 12 1 11 1 4 2
576 --- 8681 1 5 2 15 1 6 1 15 4 2 1 66 1 4 1
592 --- 51 1 30 1 5 2 4 1 205 1 6 4 4 7 4 1
608 --- 195 3 6 1 36 1 2 2 35 1 6 1 15 5 2 1
624 --- 260 15 2 2 5 1 32 1 12 2 2 1 12 2 4 2
640 --- 21541 1 4 1 9 2 4 1 757 1 10 5 4 1 6 2
656 --- 53 5 4 1 40 1 2 2 12 1 18 1 4 2 4 1
672 --- 1280 1 2 17 16 1 4 1 53 1 4 1 51 1 15 2
688 --- 42 2 8 1 5 4 2 1 44 1 2 1 36 1 62 1
704 --- 1387 1 2 1 10 1 6 4 15 1 12 2 4 1 2 1
720 --- 840 1 5 2 5 2 13 1 40 504 4 1 18 1 2 6
736 --- 195 2 10 1 15 5 4 1 54 1 2 2 11 1 39 1
752 --- 42 1 4 2 189 1 2 2 39 1 6 1 4 2 2 1
768 --- ? 1 12 1 5 1 16 4 15 5 2 1 53 1 4 5
784 --- 172 1 4 1 5 1 4 2 137 1 2 1 4 1 24 1
800 --- 1211 2 2 1 15 1 4 1 14 1 113 1 16 2 4 1
816 --- 205 1 2 11 20 1 4 1 12 5 4 1 30 1 4 2
832 --- 1630 2 6 1 9 13 2 1 186 2 2 1 4 2 10 2
848 --- 51 2 10 1 10 1 4 5 12 1 12 1 11 2 2 1
864 --- 4725 1 2 3 9 1 8 1 14 4 4 5 18 1 2 1
880 --- 221 1 68 1 15 1 2 1 61 2 4 15 4 1 4 1
896 --- 19349 2 2 1 150 1 4 7 15 2 6 1 4 2 8 1
912 --- 222 1 2 4 5 1 30 1 39 2 2 1 34 2 2 4
928 --- 235 1 18 2 5 1 2 2 222 1 4 2 11 1 6 1
944 --- 42 13 4 1 15 1 10 1 42 1 10 2 4 1 2 1
960 --- 11394 2 4 2 5 1 12 1 42 2 4 1 900 1 2 6
976 --- 51 1 6 2 34 5 2 1 46 1 4 2 11 1 30 1
992 --- 196 2 6 1 10 1 2 15 199
#############################################################################
##
#A smallgrp.g GAP group library Hans Ulrich Besche
#A & Bettina Eick
##
## This file contains the extraction function for the library of groups of
## order up to 1000 without the order 512 and 768.
##
#Y Note:
#Y All nilpotent groups have been been derived from the 2- and 3-group
#Y library of E. A. O'Brien (if possible) or have been computed using
#Y p-group generation otherwise.
#Y
#Y All soluble, non-nilpotent groups have been computed by the Frattini
#Y group extension method developed by Hans Ulrich Besche and Bettina Eick.
#Y
#Y All non-soluble groups have been computed by extending perfect groups
#Y by automorphisms.