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Transfer more properties when conjugating matrix groups #6183

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d0fe30f
Make `IsFullSubgroupGLorSLRespectingBilinearForm` docs more specific
lgoettgens Dec 22, 2025
876fd71
Transfer more properties when conjugating matrix groups
lgoettgens Dec 19, 2025
6bf6e9f
Add test
lgoettgens Dec 22, 2025
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7 changes: 5 additions & 2 deletions lib/grpmat.gd
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Original file line number Diff line number Diff line change
Expand Up @@ -345,7 +345,8 @@ DeclareAttribute( "InvariantBilinearForm", IsMatrixGroup );
## <Description>
## This property tests, whether a matrix group <A>matgrp</A> is the full
## subgroup of GL or SL (the property <Ref Prop="IsSubgroupSL"/> determines
## which it is) respecting the form stored as the value of
## which it is) in the right dimension over the (smallest) ring which
## contains all entries of its elements, respecting the form stored as the value of
## <Ref Attr="InvariantBilinearForm"/> for <A>matgrp</A>.
## </Description>
## </ManSection>
Expand Down Expand Up @@ -462,7 +463,9 @@ DeclareAttribute( "InvariantQuadraticForm", IsMatrixGroup );
## <Description>
## This property tests, whether the matrix group <A>matgrp</A> is the full
## subgroup of GL or SL (the property <Ref Prop="IsSubgroupSL"/> determines
## which it is) respecting the <Ref Attr="InvariantQuadraticForm"/> value
## which it is) in the right dimension over the (smallest) ring which
## contains all entries of its elements,
## respecting the <Ref Attr="InvariantQuadraticForm"/> value
## of <A>matgrp</A>.
## <Example><![CDATA[
## gap> g:= Sp( 2, 3 );;
Expand Down
50 changes: 50 additions & 0 deletions lib/grpmat.gi
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Expand Up @@ -1266,3 +1266,53 @@ InstallMethod( InvariantBilinearForm,
Q:= InvariantQuadraticForm( matgrp ).matrix;
return rec( matrix:= ( Q + TransposedMat( Q ) ) );
end );

#############################################################################
##
#M ConjugateGroup( <G>, <g> ) of a matrix group
##
InstallMethod( ConjugateGroup, "<G>, <g>", IsCollsElms,
[ IsMatrixGroup, IsMultiplicativeElementWithInverse ],
function( G, g )
local H, m, ginv;

H := GroupByGenerators( OnTuples( GeneratorsOfGroup( G ), g ), One(G) );
UseIsomorphismRelation( G, H );
if HasIsGeneralLinearGroup( G ) then
SetIsGeneralLinearGroup( H, IsGeneralLinearGroup( G ) );
fi;
if HasIsSpecialLinearGroup( G ) then
SetIsSpecialLinearGroup( H, IsSpecialLinearGroup( G ) );
fi;
if HasIsSubgroupSL( G ) then
SetIsSubgroupSL( H, IsSubgroupSL( G ) );
fi;
if HasInvariantBilinearForm( G ) or HasInvariantQuadraticForm( G ) then
ginv := g^-1;
fi;
if HasInvariantBilinearForm( G ) then
m := ginv * InvariantBilinearForm(G).matrix * TransposedMat(ginv);
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@fingolfin fingolfin Dec 21, 2025

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We should have test cases covering this.

I think the easiest and also most thorough way will be to modify tst/testinstall/grp/classic-forms.tst.

When it does e.g.

gap> grps:=[];;
gap> for d in [3,5,7] do
>   for q in [2,3,4,5,7,8,9,16,17,25,27] do
>     Add(grps, GO(d,q));
>   od;
> od;

the innermost loop could be changed to also add a conjugate copy of the group, i.e.

>     Add(grps, GO(d,q));
>     Add(grps, GO(d,q) ^ RandomInvertibleMat(d,GF(q));

In light of what I write in my other comment, perhaps we should also test conjugation over different fields, e.g.

Add(grps, GO(d,q) ^ RandomInvertibleMat(d,GF(q^2));

and for the unitary case deliberately over fields such as q^3 (instead of q^2)

SetInvariantBilinearForm( H, rec( matrix := m ) );
fi;
if HasInvariantQuadraticForm( G ) then
m := ginv * InvariantQuadraticForm(G).matrix * TransposedMat(ginv);
SetInvariantQuadraticForm( H, rec( matrix := m ) );
fi;
if IsSubset( FieldOfMatrixGroup( G ), FieldOfMatrixList( [ g ] ) ) then
if HasIsNaturalGL( G ) then
SetIsNaturalGL( H, IsNaturalGL( G ) );
fi;
if HasIsNaturalSL( G ) then
SetIsNaturalSL( H, IsNaturalSL( G ) );
fi;
if HasIsFullSubgroupGLorSLRespectingBilinearForm( G )
and IsFullSubgroupGLorSLRespectingBilinearForm( G ) then
SetIsFullSubgroupGLorSLRespectingBilinearForm( H, true );
fi;
if HasIsFullSubgroupGLorSLRespectingQuadraticForm( G )
and IsFullSubgroupGLorSLRespectingQuadraticForm( G ) then
SetIsFullSubgroupGLorSLRespectingQuadraticForm( H, true );
fi;
fi;
return H;
end );
20 changes: 20 additions & 0 deletions tst/testinstall/grp/classic-forms.tst
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Original file line number Diff line number Diff line change
Expand Up @@ -54,6 +54,8 @@ gap> grps:=[];;
gap> for d in [3,5,7] do
> for q in [2,3,4,5,7,8,9,16,17,25,27] do
> Add(grps, GO(d,q));
> Add(grps, GO(d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, GO(d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> od;
> od;
gap> ForAll(grps, CheckGeneratorsInvertible);
Expand All @@ -71,6 +73,10 @@ gap> for d in [2,4,6,8] do
> for q in [2,3,4,5,7,8,9,16,17,25,27] do
> Add(grps, GO(+1,d,q));
> Add(grps, GO(-1,d,q));
> Add(grps, GO(+1,d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, GO(-1,d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, GO(+1,d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> Add(grps, GO(-1,d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> od;
> od;
gap> ForAll(grps, CheckGeneratorsInvertible);
Expand All @@ -87,6 +93,8 @@ gap> grps:=[];;
gap> for d in [3,5,7] do
> for q in [2,3,4,5,7,8,9,16,17,25,27] do
> Add(grps, SO(d,q));
> Add(grps, SO(d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, SO(d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> od;
> od;
gap> ForAll(grps, CheckGeneratorsSpecial);
Expand All @@ -104,6 +112,10 @@ gap> for d in [2,4,6,8] do
> for q in [2,3,4,5,7,8,9,16,17,25,27] do
> Add(grps, SO(+1,d,q));
> Add(grps, SO(-1,d,q));
> Add(grps, SO(+1,d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, SO(-1,d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, SO(+1,d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> Add(grps, SO(-1,d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> od;
> od;
gap> ForAll(grps, CheckGeneratorsSpecial);
Expand All @@ -124,6 +136,8 @@ gap> grps:=[];;
gap> for d in [3,5,7] do
> for q in [2,3,4,5,7,8,9,16,17,25,27] do
> Add(grps, Omega(d,q));
> Add(grps, Omega(d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, Omega(d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> od;
> od;
gap> ForAll(grps, CheckGeneratorsSpecial);
Expand All @@ -141,6 +155,10 @@ gap> for d in [2,4,6,8] do
> for q in [2,3,4,5,7,8,9,16,17,25,27] do
> Add(grps, Omega(+1,d,q));
> Add(grps, Omega(-1,d,q));
> Add(grps, Omega(+1,d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, Omega(-1,d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, Omega(+1,d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> Add(grps, Omega(-1,d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> od;
> od;
gap> ForAll(grps, CheckGeneratorsSpecial);
Expand Down Expand Up @@ -191,6 +209,8 @@ gap> grps:=[];;
gap> for d in [2,4,6,8] do
> for q in [2,3,4,5,7,8,9,16,17,25,27] do
> Add(grps, Sp(d,q));
> Add(grps, Sp(d,q) ^ RandomInvertibleMat(d,GF(q)));
> Add(grps, Sp(d,q) ^ RandomInvertibleMat(d,GF(q^2)));
> od;
> od;
gap> ForAll(grps, CheckGeneratorsSpecial);
Expand Down
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