If you do SetInfoLevel(InfoLattice,,2) you can see exactly what's being
computed. Plus when I throw GAP a hard computation, I like to have some
output to give me a sense of how close it is to finishing. Along those
same lines, it would be nice if there could be a separate dialog box for
those computations so they don't end up pushing all of my input lines
off the screen, kind of like how when you install something in Ubuntu
you can open a "details" tab. I've mentioned this to Russ Woodroofe to
try to put something like it in CocoaGap.
This might be a good time to plug some computations I've done on my
homepage that give some nice presentations for the groups of order 32
and 64. I know that the Pc presentations are very fast when it comes to
algorithms, but it's hard to immediately look at the Pc presentation for
D64 with six generators and immediately see what the group is. I also
wrote a few of them as a central product.
http://www.math.wustl.edu/~bohanon/math/math.html
I can imagine it would be difficult to teach a computer what kind of
"Structure Descriptions" are the prettiest to humans. Certainly direct
products are the nicest, and then I think come central products,
semi-direct products and split extensions. Perhaps some of the methods
that James Wilson presented at CGT could eventually be used for central
products. The only problem is that for many groups there are tons of
ways to represent them as any of the last three.
Joe
Jack Schmidt wrote:
Steve Linton wrote:
A partial answer:
As you observe, these groups are all of rank 7. I think they can be
understood
as D8 x C_2^5, Q8 x C_2^5, and 2^{1+4}_+ x C_2^3 and 2^{1+4}_- x C_2^3,
although I haven't checked this fully.
Just to indicate a method as well as a solution: to guess, one can use
the simplified presentation printing from the "polycyclic" package:
gap> LoadPackage("polycyclic");;
gap> DisplayPcpGroup(Range(IsomorphismPcpGroup(SmallGroup(2^8,56083))));
...
which prints out the familiar presentation for D8 on g1,g2,g8 and then
a presentation for an elementary abelian group of order 2^5 on g3
through g7 (commutative conjugation relations are suppressed).
To verify the guess one uses a much more efficient method: construct
the direct products and use IdGroup.
IdGroup(DirectProduct(ExtraspecialGroup(2^5,"-"),ElementaryAbelianGroup(2^3)));
When I created a browsing database of the groups of order n, for n <=
2000 and NrSmallGroups(n) < 10^5, I used this method to name direct
products. It is similar to Eratosthenes sieve versus trial factoring.
The idea here is that if you are looking for direct products, then
factoring large groups is the wrong method. Rather one should form
products of small ones.
I should also mention that while the creation of the database was
enormously helpful to me, I rarely found any use of it other than a
casual reminder of "what do the groups of that order look like?"
Are there any other implementations of a computation of structure
description that gives more useful information that that in gap?
I have yet to find a convincing description for p-groups. A sparse,
algebraic presentation such as in the small groups library is about
the most useful I have found.
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