[GAP Forum] Basic question, order of fpgroup

[Stefan Kohl]

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

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