Matrix Groups over Finite Fields [HB 19]

 

New features:

• The function IsTensorInduced has been introduced which decides whether or not a matrix group is tensor-induced. It replaces and upgrades IsSymmetricTensorProduct which has been withdrawn. The related functions SymmetricTensorBasis and SymmetricTensorPermutations have been renamed TensorInducedBasis and TensorInducedPermutations respectively.

• An optional argument has been added to IsPrimitive which allows one to search for certain block sizes only; a similar argument added to IsTensor allows one to search for factors of particular dimensions.

• The function OrbitsOfSpaces(G, k) has been introduced which constructs lengths and orbit representatives for the k-dimensional spaces of the natural vector space under action of G, a matrix group defined over a prime field.

• The function NumberOfFixedSpaces(g, k) has been introduced which determines the number of k-dimensional subspaces of the natural vector space fixed under the action of $g \in GL(d, q)$.

• The function StabiliserOfSpaces has been introduced which constructs the subgroup of GL(d, q) which stabilises a sequence of subspaces of the natural vector space.

• The function EstimateOrbit(G, U), which estimates the size of the orbit of the subspace U under action of G, has been introduced.

• The function ApproximateStabiliser(G, A, U) has been introduced which constructs, using a random approach, a subgroup of the stabiliser of the subspace U under the induced action A of G.

• The permitted arguments to ExteriorSquare now include a matrix group.

• The database of irreducible soluble groups has been converted into a new style database.