`SmallGeneratingSet` for infinite groups

[ThomasBreuer]

The following happens in GAP 4.13 and in the current master branch.

gap> G:= Group( [ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 0, 1 ], [ -1, -1 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ] );;
gap> IsFinite( G );
false
gap> SmallGeneratingSet( G );
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 3rd choice method found for `Enumerator' on 1 arguments at /.../gap/lib/methsel2.g:250 called from
[...]

The method used for this infinite group is SMALLGENERATINGSETGENERIC, which tries to omit generators from the stored list. For that, IsSubset gets called, which does membership tests. And there is no suitable method for that.

What would be a good strategy to decide whether calling SmallGeneratingSet with an infinite group will run into an error? For which (nontrivial) situations with infinite groups does SMALLGENERATINGSETGENERIC something useful?

[Stefan-Kohl]

@ThomasBreuer Here is a nontrivial example of where SMALLGENERATINGSETGENERIC does something useful for an infinite group (the example uses the RCWA package):

gap> a := ClassTransposition(0,2,1,2);;
gap> b := ClassTransposition(1,2,2,4);;
gap> c := ClassTransposition(1,4,2,6);;
gap> d := ClassTransposition(0,4,3,6);;
gap> e := ClassTransposition(1,4,7,12);;
gap> G := Group(a,b,c,d,e);
<(0(2),1(2)),(1(2),2(4)),(1(4),2(6)),(0(4),3(6)),(1(4),7(12))>
gap> SMALLGENERATINGSETGENERIC(G);
[ ( 0(2), 1(2) ), ( 1(2), 2(4) ), ( 0(4), 3(6) ) ]
gap> time;
316
gap> G = Group(SMALLGENERATINGSETGENERIC(G)); 
true

(The example is nontrivial insofar as the group G acts transitively on the set of nonnegative integers if and only if the Collatz conjecture holds.)

[ThomasBreuer]

Thanks for the example.

Meanwhile I noticed that this issue is a duplicate of #5542.

[fingolfin]

Was resolved by PR #5795