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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346# Circle, chapter 3 # [ ".//doc/funct.xml", 41, 46 ] gap> a := CircleObject( 2 ); CircleObject( 2 ) # [ ".//doc/funct.xml", 57, 64 ] gap> a := CircleObject( 2 ); CircleObject( 2 ) gap> UnderlyingRingElement( a ); 2 # [ ".//doc/funct.xml", 87, 97 ] gap> IsCircleObject( 2 ); IsCircleObject( CircleObject( 2 ) ); false true gap> IsMultiplicativeElementWithInverse( CircleObject( 2 ) ); true gap> IsCircleObjectCollection( [ CircleObject(0), CircleObject(2) ] ); true # [ ".//doc/funct.xml", 117, 124 ] gap> IsPositionalObjectOneSlotRep( CircleObject( 2 ) ); true gap> IsDefaultCircleObject( CircleObject( 2 ) ); true # [ ".//doc/funct.xml", 140, 145 ] gap> FamilyObj( CircleObject ( 2 ) ) = CircleFamily( FamilyObj( 2 ) ); true # [ ".//doc/funct.xml", 164, 173 ] gap> One( CircleObject( 5 ) ); CircleObject( 0 ) gap> One( CircleObject( 5 ) ) = CircleObject( Zero( 5 ) ); true gap> One( CircleObject( [ [ 1, 1 ],[ 0, 1 ] ] ) ); CircleObject( [ [ 0, 0 ], [ 0, 0 ] ] ) # [ ".//doc/funct.xml", 191, 200 ] gap> CircleObject( -2 )^-1; CircleObject( -2 ) gap> CircleObject( 2 )^-1; CircleObject( -2/3 ) gap> CircleObject( -2 )*CircleObject( -2 )^-1; CircleObject( 0 ) # [ ".//doc/funct.xml", 202, 213 ] gap> m := CircleObject( [ [ 1, 1 ], [ 0, 1 ] ] ); CircleObject( [ [ 1, 1 ], [ 0, 1 ] ] ) gap> m^-1; CircleObject( [ [ -1/2, -1/4 ], [ 0, -1/2 ] ] ) gap> m * m^-1; CircleObject( [ [ 0, 0 ], [ 0, 0 ] ] ) gap> CircleObject( [ [ 0, 1 ], [ 1, 0 ] ] )^-1; fail # [ ".//doc/funct.xml", 228, 242 ] gap> IsUnit( Integers, CircleObject( -2 ) ); true gap> IsUnit( Integers, CircleObject( 2 ) ); false gap> IsUnit( Rationals, CircleObject( 2 ) ); true gap> IsUnit( ZmodnZ(8), CircleObject( ZmodnZObj(2,8) ) ); true gap> m := CircleObject( [ [ 1, 1 ],[ 0, 1 ] ] );; gap> IsUnit( FullMatrixAlgebra( Rationals, 2 ), m ); true # [ ".//doc/funct.xml", 249, 260 ] gap> IsUnit( CircleObject( -2 ) ); true gap> IsUnit( CircleObject( 2 ) ); false gap> IsUnit( CircleObject( ZmodnZObj(2,8) ) ); true gap> IsUnit( CircleObject( [ [ 1, 1 ],[ 0, 1 ] ] ) ); false # [ ".//doc/funct.xml", 276, 291 ] gap> IsCircleUnit( Integers, -2 ); true gap> IsCircleUnit( Integers, 2 ); false gap> IsCircleUnit( Rationals, 2 ); true gap> IsCircleUnit( ZmodnZ(8), ZmodnZObj(2,8) ); true gap> m := [ [ 1, 1 ],[ 0, 1 ] ]; [ [ 1, 1 ], [ 0, 1 ] ] gap> IsCircleUnit( FullMatrixAlgebra(Rationals,2), m ); true # [ ".//doc/funct.xml", 296, 307 ] gap> IsCircleUnit( -2 ); true gap> IsCircleUnit( 2 ); false gap> IsCircleUnit( ZmodnZObj(2,8) ); true gap> IsCircleUnit( [ [ 1, 1 ],[ 0, 1 ] ] ); false # [ ".//doc/funct.xml", 331, 338 ] gap> R:=Ring( [ ZmodnZObj(2,8) ] ); <ring with 1 generators> gap> S:=AdjointSemigroup(R); <monoid with 4 generators> # [ ".//doc/funct.xml", 391, 408 ] gap> x:=[ [ 0, 1, 0 ], > [ 0, 0, 1 ], > [ 0, 0, 0 ] ];; gap> R := Algebra( GF(2), [ One(GF(2))*x ] ); <algebra over GF(2), with 1 generators> gap> RadicalOfAlgebra( R ) = R; true gap> Dimension(R); 2 gap> G := AdjointGroup( R );; gap> Size( R ) = Size( G ); true gap> StructureDescription( G ); "C4" # [ ".//doc/funct.xml", 417, 436 ] gap> x:=[ [ 0, 1, 0 ], > [ 0, 0, 0 ], > [ 0, 0, 0 ] ];; gap> y:=[ [ 0, 0, 0 ], > [ 0, 0, 1 ], > [ 0, 0, 0 ] ];; gap> R := Algebra( GF(2), One(GF(2))*[x,y] ); <algebra over GF(2), with 2 generators> gap> RadicalOfAlgebra(R) = R; true gap> Dimension(R); 3 gap> G := AdjointGroup( R ); <group of size 8 with 2 generators> gap> StructureDescription( G ); "D8" # [ ".//doc/funct.xml", 452, 466 ] gap> R := Ring( [ ZmodnZObj(2,8) ] ); <ring with 1 generators> gap> G := AdjointGroup( R ); <group of size 4 with 2 generators> gap> StructureDescription( G ); "C2 x C2" gap> R := Ring( [ ZmodnZObj(2,256) ] ); <ring with 1 generators> gap> G := AdjointGroup( R );; gap> StructureDescription( G ); "C64 x C2" # [ ".//doc/funct.xml", 474, 487 ] gap> R := Ring( [ ZmodnZObj(2,256) ] ); <ring with 1 generators> gap> S := AdjointSemigroup( R ); <monoid with 128 generators> gap> H := GreensHClassOfElement(S,One(S)); <Green's H-class: <object>> gap> G:=AsGroup(H); <group of size 128 with 2 generators> gap> StructureDescription(G); "C64 x C2" # [ ".//doc/funct.xml", 497, 509 ] gap> R := Ring( [ ZmodnZObj(2,256) ] ); <ring with 1 generators> gap> AdjointGroup(R);; gap> R := Ring( [ ZmodnZObj(2,256) ] ); <ring with 1 generators> gap> S:=AdjointSemigroup(R); <monoid with 128 generators> gap> AsGroup(GreensHClassOfElement(S,One(S))); <group of size 128 with 2 generators> # [ ".//doc/funct.xml", 528, 542 ] gap> LoadPackage( "laguna", false ); true gap> FG := GroupRing( GF(2), DihedralGroup(8) ); <algebra-with-one over GF(2), with 3 generators> gap> R := AugmentationIdeal( FG );; gap> G := AdjointGroup( R );; gap> IdGroup( G ); [ 128, 170 ] gap> IdGroup( Units( FG ) ); #I LAGUNA package: Computing the unit group ... [ 128, 170 ] # [ ".//doc/funct.xml", 551, 568 ] gap> Size( AdjointGroup( GroupRing( GF(2), DihedralGroup(8) ) ) ); WARNING: usage of AdjointGroup for associative ring <R> with one!!! In this case the adjoint group is isomorphic to the unit group Units(<R>), which possibly may be computed faster!!! 128 gap> Size( AdjointGroup( Integers mod 11 ) ); WARNING: usage of AdjointGroup for associative ring <R> with one!!! In this case the adjoint group is isomorphic to the unit group Units(<R>), which possibly may be computed faster!!! 10 # [ ".//doc/funct.xml", 593, 605 ] gap> SetInfoLevel( InfoCircle, 1 ); gap> SetInfoLevel(InfoCircle,1); gap> R := Ring( [ ZmodnZObj(2,8) ]); <ring with 1 generators> gap> G := AdjointGroup( R ); #I Circle : <R> is not a radical algebra, computing circle units ... #I Circle : searching generators for adjoint group ... <group of size 4 with 2 generators> gap> SetInfoLevel( InfoCircle, 0 );