Dear Stefan, Thanks very much. That's very helpful.
That it's infinite is very surprising. I believed G to be an extension of S_5 by a finite product of cyclic groups of order 2. I need to think a lot more about this. Best, Kirill On 27 July 2011 17:22, Stefan Kohl <ste...@mcs.st-and.ac.uk> wrote: > Dear Forum, > > Kirill Mackenzie asked: > > [ ... ] > > > f:=FreeGroup(4);; > > G:=f/[f.1^2,f.2^2,f.3^2,f.4^2, > > (f.1*f.2)^3,(f.1*f.3)^3,(f.1*f.4)^3,(f.2*f.3)^3,(f.2*f.4)^3,(f.3*f.4)^3, > > (f.1*f.2*f.1*f.3)^4,(f.1*f.2*f.1*f.4)^4, > > (f.1*f.3*f.1*f.4)^4,(f.1*f.3*f.1*f.2)^4, > > (f.1*f.4*f.1*f.2)^4,(f.1*f.4*f.1*f.3)^4, > > (f.2*f.3*f.2*f.4)^4,(f.2*f.3*f.2*f.1)^4, > > (f.2*f.4*f.2*f.1)^4,(f.2*f.4*f.2*f.3)^4, > > (f.2*f.1*f.2*f.3)^4,(f.2*f.1*f.2*f.4)^4, > > (f.3*f.4*f.3*f.2)^4,(f.3*f.4*f.3*f.1)^4, > > (f.3*f.1*f.3*f.2)^4,(f.3*f.1*f.3*f.4)^4, > > (f.3*f.2*f.3*f.1)^4,(f.3*f.2*f.3*f.4)^4, > > (f.4*f.1*f.4*f.2)^4,(f.4*f.1*f.4*f.3)^4, > > (f.4*f.2*f.4*f.1)^4,(f.4*f.2*f.4*f.3)^4, > > (f.4*f.3*f.4*f.2)^4,(f.4*f.3*f.4*f.1)^4]; > > > To start I want the order of G. With `Size(G);' `Order(G);' > `IsFinite(G);' > > I just get several (8 or 9) iterations of > > > > #I Coset table calculation failed -- trying with bigger table limit > > > > and then > > > > exceeded the permitted memory (`-o' command line option) at > > A common strategy to decide finiteness of an fp group which is > 'often' successful is to search for low-index subgroups which > have the infinite cyclic group as an homomorphic image, i.e. > which have 0 among their abelian invariants: > > gap> low := LowIndexSubgroupsFpGroup(G,20);; > gap> Set(List(low,AbelianInvariants)); > [ [ ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 2 ], [ 2 ], [ 2, 2 ], [ 2, 2, 2, 2 ], > [ 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 3 ], [ 2, 2, 2, 2, 4 ], [ 2, 2, 2, 4 ], > [ 2, 2, 4 ], [ 2, 2, 8 ], [ 2, 4, 4, 4 ], [ 3 ] ] > > >From this you see that your group G has such subgroups, > thus it is infinite. > > Further you can compute finite quotients of your group > by letting it act by multiplication from the right on the > right cosets of a subgroup. -- For example: > > gap> quots := List(low,H->Action(G,RightCosets(G,H),OnRight));; > gap> List(quots,Size); > [ 1, 120, 1920, 1920, 2432902008176640000, 1920, 1920, 1920, > 2432902008176640000, 1920, 1920, 2432902008176640000, 1920, 1920, 1920, > 1920, 1920, 1920, 2432902008176640000, 1920, 1920, 2432902008176640000, > 1920, 1920, 1920, 1920, 120, 3840, 3840, 3840, 3840, 3840, 3840, 3840, > 1920, 1920, 1920, 1920, 1920, 3840, 3840, 2432902008176640000, 3840, 3840, > 3840, 3840, 3840, 120, 1920, 1920, 1920, 1920, 120, 3840, 3840, 3840, > 3840, > 3840, 3840, 3840, 120, 2, 120, 120, 3840, 3840, 3840, 3840, 3840, 3840, > 3840, 120, 3840, 95040, 3840, 3840, 239500800, 3840, 3840, 95040, 3840, > 3840, 3840, 3840, 95040, 3840, 239500800, 239500800, 3840, 95040, 3840, > 95040, 3840, 239500800, 3840, 95040, 120 ] > > For example you see that your group has a quotient which is > isomorphic to the Mathieu group M12: > > gap> List(Filtered(quots,IsSimple),StructureDescription); > [ "C2", "M12", "A12", "M12", "M12", "A12", "A12", "M12", "M12", "A12", > "M12" ] > > Hope this helps, > > Stefan Kohl > > --------------------------------------------------------------------------- > http://www.gap-system.org/DevelopersPages/StefanKohl/ > --------------------------------------------------------------------------- > > > > _______________________________________________ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum > _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
