Revise tests to check forms directly

[fingolfin]

Testing IsSubset(G, elm) can be slow in GAP if G is a matrix group of "large" degree. This is because GAP resorts to testing membership using a stabilizer chain in an isomorphic permutation group, even if G is a linear/unitary/orthogonal/symplectic group. So we replace that check with one that actually tests using the invariant forms of the groups in question.

This is work in progress, and waiting for PR #88 and needs a few functions such as SpIsotropicImprimitives to set invariant forms -- which PR #89 seems to do.

Checklist for the reviewer

General

• [ ] All functions have unit tests

Functions constructing generators of maximal subgroups

• [ ] The code which computes and then stores the sizes is correct. To do this one confers to the tables referenced by the code. Tests for the group size should use RECOG.TestGroup from the recog package.
• [ ] The test suite checks whether all constructed generators are sensible. That is it tests the generators'
  • [ ] determinants (and if applicable also the spinor norms are correct), and
  • [ ] compatibility with the correct form,
  • [ ] DefaultFieldOfMatrixGroup returns the correct field.

Functions assembling the list of all maximal subgroups of a certain group

• [ ] The code agrees with the tables in section 8.2 of the book "The Maximal Subgroups of the Low-dimensional Finite Classical Groups".

The reviewer doesn't need to compare our results to magma's results. That's the job of the person implementing the code.

[fingolfin]

@TristanPfersdorff these tests caught one omission in SubfieldSp (it did not always set the invariant form, fixed in this PR), and potentially also a "real" bug in SymplecticSubfieldSU: for inputs n=2,q=4 the returned group does not preserve the given sesquilinear form. Indeed, according to a brute force search, it doesn't seem to preserve any such form:

gap> G:=SymplecticSubfieldSU(2,4);;
gap> Filtered(GF(4)^[2,2], M -> ForAll(GeneratorsOfGroup(G), g -> g*M*HermitianConjugate(g,2) = M));
[ [ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ] ]

[fingolfin]

Oh no wait, I misunderstood the function arguments, the base field is GF(4) and the big field is GF(16). OK, so trying again:

gap> G:=SymplecticSubfieldSU(2,4);;
gap> forms:=Filtered(GF(4)^[2,2], M -> ForAll(GeneratorsOfGroup(G), g -> g*M*HermitianConjugate(g,4) = M));
[ [ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ], [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ],
  [ [ 0*Z(2), Z(2^2) ], [ Z(2^2), 0*Z(2) ] ], [ [ 0*Z(2), Z(2^2)^2 ], [ Z(2^2)^2, 0*Z(2) ] ] ]
gap> Display(forms[2]);
 . 1
 1 .
gap> Display(InvariantSesquilinearForm(G).matrix);
 . 1
 1 .
gap> CheckSesquilinearForm(G);
false

The bug was in CheckSesquilinearForm, fixed now.

[codecov[bot]]

Codecov Report

Merging #91 (b40ffd9) into main (9365a36) will not change coverage. The diff coverage is 100.00%.

@@           Coverage Diff           @@
##             main      #91   +/-   ##
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  Coverage   96.15%   96.15%           
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  Files          14       14           
  Lines        2734     2734           
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  Hits         2629     2629           
  Misses        105      105           

Impacted Files	Coverage Δ
gap/SubfieldMatrixGroups.gi	97.82% <100.00%> (ø)