[GAP Forum] Forming only groups smaller than a certain size
Alexander Hulpke
hulpke at math.colostate.edu
Tue Jul 20 20:20:27 BST 2010
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
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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