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#@local a4, k4, rc, lc, a, b, d, f, g, h, H1, H2, H3, K1, K2, K3, lc1, lc4, M, matcyc, rc5, s
#############################################################################
## adapted from gapdev/tst/tstinstall/cset.tst for left cosets
##
## test of group intersection and LeftCoset
##
gap> START_TEST("lcset.tst");
# section 5.2.1: basic coset tests
gap> a4 := Group( (1,2,3), (2,3,4) );; SetName( a4, "a4" );
gap> k4 := Group( (1,2)(3,4), (1,3)(2,4) );; SetName( k4, "k4" );
gap> rc := RightCosets( a4, k4 );
[ RightCoset(k4,()), RightCoset(k4,(2,3,4)), RightCoset(k4,(2,4,3)) ]
gap> lc := LeftCosets( a4, k4 );
[ LeftCoset((),k4), LeftCoset((2,4,3),k4), LeftCoset((2,3,4),k4) ]
gap> AsSet( lc[2] );
[ (2,4,3), (1,2,3), (1,3,4), (1,4,2) ]
gap> LeftCoset( (1,4,2), k4 ) = lc[2];
true
gap> Representative( lc[2] );
(2,4,3)
gap> ActingDomain( lc[2] );
k4
gap> (1,4,3) in lc[3];
true
gap> (1,2,3)*lc[2] = lc[3];
true
gap> lc[2]^(1,3,2) = lc[3];
true
# section 5.2.2
gap> Inverse( rc[3] ) = lc[3];
true
gap> Inverse( lc[2] ) = rc[2];
true
# many further tests
gap> LeftCoset( (1,2), a4 ) = LeftCoset( (2,3), a4 );
true
gap> () in LeftCoset( (1,2), Group([(1,2,3,4)]) );
false
gap> (1,2) in LeftCoset( (5,6), SymmetricGroup(12) );
true
gap> Length( LeftCosets( SymmetricGroup(5), AlternatingGroup(4) ) );
10
gap> (1,2,3) * LeftCoset( (), AlternatingGroup(4) )
> = LeftCoset( (), AlternatingGroup(4) );
true
gap> IsBiCoset( LeftCoset( (1,2), AlternatingGroup(6) ) );
true
gap> IsBiCoset( LeftCoset( (1,7), AlternatingGroup(6) ) );
false
gap> IsLeftCoset( LeftCoset( (1,2,3), MathieuGroup(12) ) );
true
gap> g:=SymmetricGroup(3);;
gap> h:=Group((1,2));;
gap> List(LeftCosets(g,h), SSortedList);
[ [ (), (1,2) ], [ (1,3,2), (1,3) ], [ (2,3), (1,2,3) ] ]
# test intersecting permutation cosets
gap> H1 := Group( [ (), (2,7,6)(3,4,5), (1,2,7,5,6,4,3) ] );;
gap> H2 := Group( [ (1,2,3,4,5,6,7), (5,6,7) ] );;
gap> H3 := Group( [ (1,2,3,4,5,6,8), (1,3,2,6,4,5), (1,6)(2,3)(4,5)(7,8) ] );;
gap> AsSet( LeftCoset( (1,5,7,3)(4,6), H1 ) ) =
> Intersection( LeftCoset( (3,6)(4,7), H2 ),
> LeftCoset( (1,5,3,8,6,7), H3 ) );
true
gap> AsSet( LeftCoset( (1,2,5,6,7,4,3,8), Group(()) ) ) =
> Intersection( LeftCoset( (1,5,6,7)(3,8,4),
> Group( [ (1,4)(2,5), (1,3,5)(2,4,6), (1,5)(2,4)(3,6) ] ) ),
> LeftCoset( (1,2,6,8)(3,7),
> Group( [ (3,4), (5,6,7,8), (5,6) ] ) ) );
true
gap> [] = Intersection( LeftCoset( (), SymmetricGroup(4) ),
> LeftCoset( (4,7), SymmetricGroup([3..6]) ) );
true
gap> [] = Intersection( LeftCoset( (4,5), Group( [ (1,2,3,4,5) ] ) ),
> LeftCoset( (), AlternatingGroup(4) ) );
true
gap> AsSet( LeftCoset( (7,9), SymmetricGroup([3..5]) ) ) =
> Intersection( LeftCoset( (1,2)(7,9), SymmetricGroup(5) ),
> LeftCoset( (7,9), SymmetricGroup([3..7]) ) );
true
gap> [] = Intersection( LeftCoset( (1,4)(3,5), Group([(1,2,3,4,5)]) ),
> LeftCoset( (), SymmetricGroup(3) ) );
true
gap> AsSet( LeftCoset( (4,5), Group( [(5,6)] ) ) ) =
> Intersection( LeftCoset( (), SymmetricGroup(6) ),
> LeftCoset( (4,5), SymmetricGroup([5..8]) ) );
true
gap> AsSet( LeftCoset( (1,4,5), SymmetricGroup(5) ) ) =
> Intersection( LeftCoset( (), SymmetricGroup(5) ),
> LeftCoset( (1,2), SymmetricGroup(5) ) );
true
gap> [] =
> Intersection( LeftCoset( (1,2), Group( (1,2,3,4,5) ) ),
> LeftCoset( (), Group( (1,2,3,5,4) ) ) );
true
gap> [] =
> Intersection( LeftCoset( (1,2,3), Group( (1,2,3,4,5) ) ),
> LeftCoset( (), Group( (1,2,3,5,4) ) ) );
true
gap> AsSet( LeftCoset( (1,2), Group( [ (1,2,3,5,4) ] ) ) ) =
> Intersection( LeftCoset( (), SymmetricGroup(7) ),
> LeftCoset( (1,2), Group( (1,2,3,5,4) ) ) );
true
gap> [] =
> Intersection( LeftCoset( (), SymmetricGroup([3..7]) ),
> LeftCoset( (1,2), Group( (1,2,3,5,4) ) ) );
true
# test trivial cases
gap> Intersection( LeftCoset( (), Group([],()) ),
> LeftCoset( (1,2), Group([],()) ) ) = [];
true
gap> Intersection( LeftCoset( (), Group( (1,2,3) ) ),
> LeftCoset( (1,2), Group( (1,2,3) ) ) ) = [];
true
gap> Intersection( LeftCoset( (), AlternatingGroup(6) ),
> LeftCoset( (1,2), AlternatingGroup(6) ) ) = [];
true
gap> Intersection( LeftCoset( (1,2), AlternatingGroup([1..5]) ),
> LeftCoset( (1,2), AlternatingGroup([6..10]) ) )
> = AsSet( LeftCoset( (1,2), Group(()) ) );
true
#coset of pc-group
gap> d := DihedralGroup( 24 );
<pc group of size 24 with 4 generators>
gap> List( GeneratorsOfGroup(d), x -> Order(x) );
[ 2, 12, 6, 3 ]
gap> s := Subgroup( d, [ d.1, d.4 ] );;
gap> SetName( s, "s" );
gap> lc4 := LeftCoset( d.2, s );
LeftCoset(<object>,s)
gap> AsSet( lc4 );
[ f2, f2*f4, f1*f2*f3, f2*f4^2, f1*f2*f3*f4, f1*f2*f3*f4^2 ]
gap> d.2 * d.4 in lc4;
true
# coset of fp-group
gap> f := FreeGroup(2);; a := f.1;; b := f.2;;
gap> g := f / [ a^5, b^4, a*b*a^2*b^3 ];
<fp group on the generators [ f1, f2 ]>
gap> Size(g);
20
gap> h := Subgroup( g, [g.1] );;
gap> SetName( h, "C5" );
gap> rc5 := LeftCoset( g.2, h );
LeftCoset(<object>,C5)
gap> AsSet( rc5 );
[ f2, f2*f1, f2*f1^2, f2*f1^3, f2*f1^4 ]
# test intersection non-permutation cosets
gap> K1 := Group( [ [[-1,0],[0,-1]] ] );;
gap> K2 := Group( [ [[-1,0],[0,1]], [[0,1],[1,0]] ] );;
gap> K3 := Group( [ - IdentityMat(2) ] );;
gap> AsSet( LeftCoset( [ [0,1],[1,0] ], K1 ) ) =
> Intersection( LeftCoset( IdentityMat(2), K2 ),
> LeftCoset( [[0,1],[1,0]], K1 ) );
true
gap> AsSet( LeftCoset( [[0,1],[1,0]], K3 ) ) =
> Intersection( LeftCoset( [[0,-1],[-1,0]], K3 ),
> LeftCoset( [[0,1],[1,0]], K3 ) );
true
gap> matcyc := CyclicGroup( IsMatrixGroup, GF(3), 4 );;
gap> M := GeneratorsOfGroup( matcyc )[1];;
gap> lc1 := LeftCoset( M^2, matcyc );;
gap> Representative(lc1);
[ [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ],
[ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ] ]
#
gap> STOP_TEST("lcset.tst", 1);
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