[GAP Forum] Forming only groups smaller than a certain size

[Alexander Hulpke]

Dear Forum, Dear Krishna Mohan,

>     I am using matrices as generators. The set of generators consist of just 2 elements. Each element by itself generates a small group. 

If you have ``random'' matrices the lartge groups you want to exclude likely are very large (containing SL). As a quick way of eliminating such giants, you could (even before trying an orbit calculation) try the orders of a few (pseudo-)random elements (you could replace 20 by a different number):

List([1..20],x->Order(PseudoRandom(G)));

and calculate the LCM. This is a lower bound for the group order which will typically eliminate very large subgroups of GL very quickly.

Also (assuming you're working over a finite field, you could look at the action on submodules or factor modules first: (again they give lower bounds on the order):

m:=GModuleByMats(GeneratorsOfGroup(G),DefaultFieldOfMatrixGroup(G));
MTX.IsIrreducible(m);
if this returns false, you can use:

bas:=MTX.ProperSubmoduleBasis(m);
l:=MTX.InducedAction(m,bas);

Now l[1].generators gives matrices (in smaller dimension) for the action on the submodule,
l[2].generators for the action on the factor module. The orders of the corresponding groups again are lower bounds for |G|.

Best,

   Alexander Hulpke




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