Coset Enumeration with GAP

Question

I have downloaded GAP version 4.7.8. for windows, and installed everything (all packages, including"ACE") with the installer. Now I want to do a simple task, enumerating the cosets. To create a group and a subgroup I execute:

gap> G := SymmetricGroup( 8 );
Sym( [ 1 .. 8 ] )
gap> U := Subgroup( G, [ (1,2), (3,4), (3,4,5) ] );
Group([ (1,2), (3,4), (3,4,5) ])

which works fine. Now for coset enumeration I found in the GAP Manual for Version 4.7.8 the function CosetTable, but when I trying to execute in analogy with the example given there I get:

gap> tab := CosetTable(G, U);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `CosetTable' on 2 arguments called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
called from read-eval loop at line 3 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue

which leaves me with a big question sign? Using "? Coset Table" directly in GAP just gives me also the above section in the manual. Now looking further through the manual I found the function "CosetTableBySubgroup", executing gives me

gap> CosetTableBySubgroup(G,U);
[ [ 2, 4, 1, 6, 8, 3, 5, 10, 7, 9 ], [ 3, 1, 6, 2, 7, 4, 9, 5, 10, 8 ], 
  [ 1, 5, 7, 4, 2, 6, 3, 9, 8, 10 ], [ 1, 5, 7, 4, 2, 6, 3, 9, 8, 10 ] ]

So luckily something that works, but what would be preferable would be a list of cosets, something like this GAP Code which unluckily does not work (maybe because it refers to an old version), because it gives:

gap> LeftCosets(G, U);
Error, Variable: 'LeftCosets' must have a value not in any function at line 6 of *stdin*

While searching through the web I found this document, stating that you have to define the function "CosetTable" first, but when I try to execute the code there I get:

gap> CosetTable:=function(g,n)
> local x,y,tmp;
> tmp:=Flat(List(LeftCosets(g,n),x->Elements(x)));
Syntax error: warning: unbound global variable
tmp:=Flat(List(LeftCosets(g,n),x->Elements(x)));
                     ^

So does not help. Then I found the package "ACE", and tried to use it, but this package, despite being installed, does not load on my system:

gap> LoadPackage("ace");
#I  Package ``ACE'': The program `ace' is not compiled
fail

How should I compile, I have no compiler on my Windows 8.1 system, and I used the installer, which should have all packages pre-installed on it?

So how can I enumerate the cosets in GAP?????

Answer 1

Forget about the external binaries (or the ACE package, or in fact any packages at all) at the moment -- they can give runtime improvements, but that is not what you need here. Also the web pages you link are not the best references. Go to http://www.gap-system.org.

CosetTable or "coset enumeration" refers to a technical concept (a particular way to write down a permutation representation for finitely presented groups. Again this is most likely not what you are after. (The CosetTableBySubgroup gives such a table -- each coset is represented by a number.)

If you want a list of the cosets (GAP uses right cosets, not left cosets and there is basically no functionality for left cosets), you can use the command RightCosets:

gap> g:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> s:=Subgroup(g,[(1,2,3),(2,3,4)]);
Group([ (1,2,3), (2,3,4) ])
gap> SetName(s,"mysubgroup");
gap> cosets:=RightCosets(g,s);
[ RightCoset(mysubgroup,()), RightCoset(mysubgroup,(3,4)),
  RightCoset(mysubgroup,(1,5)), RightCoset(mysubgroup,(1,5)(3,4)),
  RightCoset(mysubgroup,(1,5,2)), RightCoset(mysubgroup,(1,5,2)(3,4)),
  RightCoset(mysubgroup,(1,5,3)), RightCoset(mysubgroup,(1,5,3,4)),
  RightCoset(mysubgroup,(1,5,4)), RightCoset(mysubgroup,(1,5,4,3)) ]

You can then determine the action on the cosets by multiplication

gap> act:=Action(g,cosets,OnRight);
Group([ (1,4,5,7,9)(2,3,6,8,10), (1,2)(3,5)(4,6)(7,8)(9,10) ])

or ActionHomomorphism (which will not return the image group, but a homomorphism.