Transfer more information when conjugating a matrix group

[danielrademacher]

Observed behaviour

Taking e.g.

g := PseudoRandom(GL(100,5));
G := Sp(100,5)^g;

we can ask

h := PseudoRandom(GL(100,5));
h in G;

which takes extremely long. GAP knows some information about G, .e.g. the size since it knows the size of Sp(100,5), but GAP could also know that G is a group preserved by a form. It also knows the form since Sp(100,5) is a group preserved by a form.

Expected behaviour

Writing

G := Sp(100,5)^g;

is probably supposed to be very fast, but I think it would be nice to have some functionality (maybe by calling another function) such that GAP transfers more information.

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[fingolfin]

This should be a matter of adding a new method to ConjugateGroup for matrix groups, which takes care of transferring some extra information, in particular the properties & attributes...

• IsFullSubgroupGLorSLRespectingBilinearForm
  • BTW this really should be a plain filter not a property
• IsFullSubgroupGLorSLRespectingSesquilinearForm
• IsFullSubgroupGLorSLRespectingQuadraticForm
• InvariantBilinearForm (after adjusting it suitably of course, i.e. conjugating the matrix)
• InvariantSesquilinearForm (ditto)
• InvariantQuadraticForm (ditto)

Right now the generic method is

InstallMethod( ConjugateGroup, "<G>, <g>", IsCollsElms,
    [ IsGroup, IsMultiplicativeElementWithInverse ],
    function( G, g )
    local   H;

    H := GroupByGenerators( OnTuples( GeneratorsOfGroup( G ), g ), One(G) );
    UseIsomorphismRelation( G, H );
    return H;
end );

which could become

InstallMethod( ConjugateGroup, "<G>, <g>", IsCollsElms,
    [ IsMatrixGroup, IsMultiplicativeElementWithInverse ],
    function( G, g )
    local   H, m;

    H := GroupByGenerators( OnTuples( GeneratorsOfGroup( G ), g ), One(G) );
    UseIsomorphismRelation( G, H );
    if HasInvariantBilinearForm( G ) then
      m := TransposedMat(g) * InvariantBilinearForm(G).matrix * g;
      SetInvariantBilinearForm( H, rec( matrix := m ) );
    fi;
    ... TODO add more
    return H;
end );