From: rusin@teton.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Groups and my software
Date: 12 Oct 1997 19:46:19 GMT
In article <61mtu0$o38@freenet-news.carleton.ca>,
Paul Derbyshire wrote:
[he is writing code to enumerate finite groups of small order by
examining their multiplication tables.]
>It got so slow at order 9 I ported it to C++. Now strangely it crashes at
>order 7, seems the compiler's "new" operator code is buggy since that's
>where it segs out, but it does the others fine including 8 and finds all
>5. Now for order 9 it found just two: Z_9 and Z_3xZ_3. Is it missing any? ;-)
The groups of low order are, up to isomorphism,
Order Group
1 C_1
2 C_2
3 C_3
4 C_4, C_2 x C_2
5 C_5
6 C_6, S_3
7 C_7
8 C_8, C_2 x C_4, C_2 x C_2 x C_2, D_8, Q_8
9 C_9, C_3 x C_3
10 C_10, D_10
11 C_11
12 C_12, C_2 x C_6, D_12, A_4, X
13 C_13
14 C_14, D_14
15 C_15
(C_n is cyclic of order n, D_n is dihedral of order n, Q_n is the quaternion
group of order n. S_n is the symmetric group on n letters, and A_n is the
alternating group on n letters.) The group X is a semidirect product of
C_3 and C_4, the latter acting on the former by inversion.
I hate to discourage anyone working on a mathematical project, but I have
to say that enumerating groups by creating lots of Cayley tables and then
searching for isomorphisms among them is really kind of pointless. This
tends to treat the associative law as sort of a nice accident when it happens,
rather than a key feature of group theory. It is tremdously easier and
more satisfying to classify groups of low order by invoking the Sylow
theorems, group actions, and cohomology. (In fact, this technique is in
principle applicable to the classification of solvable groups of any order.
It's a little less direct when looking for simple groups...)
You should also know that there are really very many groups of some orders.
Why, you could even write a book on The Groups of Order 2^n, n <= 6. (hint)
dave
(See also:
http://www.math.niu.edu/~rusin/known-math/95/numgrps
http://www.math.niu.edu/~rusin/known-math/97/numgrps
for a few related comments.)
==============================================================================
[URLs updated 1999/01 -- djr]