[GAP Forum] StructureDescription & memory

[Joe Bohanon]

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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