64.2 IsFinite for rational matrix groups

IsFinite( G )

The group G, which must consist of rational matrices, is tested for finiteness.

A group of rational matrices is finite iff the following two conditions hold: There is a basis with respect to which all elements of G have integer entries, and G preserves a positive definite quadratic form.

If G contains non-integer matrices, then IsFinite first calls InvariantLattice (see InvariantLattice for rational matrix groups) to find a basis with respect to which all elements of G are integer matrices.

IsFinite then finds a positive definite quadratic form, or determines that none exists. If G is finite, then the quadratic form is stored in G.quadraticForm.

gap> a := [[1,1/2],[0,-1]];; G := Group(a);;
gap> IsFinite(G);
true
gap> L := G.invariantLattice;;
gap> L*a*L^(-1);
[ [ 1, 1 ], [ 0, -1 ] ]
gap> B := G.quadraticForm;
[ [ 4, 1 ], [ 1, 3/2 ] ]
gap> TransposedMat(a)*B*a;
[ [ 4, 1 ], [ 1, 3/2 ] ]

This function is Las Vegas: it is randomized, but the randomness only affects the running time, not the correctness of the output. (See IsFiniteDeterministic for integer matrix groups.)

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