# utils, chapter 5
#
# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!
#
# This file has been generated by AutoDoc. It contains examples extracted from
# the package documentation. Each example is preceded by a comment which gives
# the name of a GAPDoc XML file and a line range from which the example were
# taken. Note that the XML file in turn may have been generated by AutoDoc
# from some other input.
#
gap> START_TEST("utils05.tst");

# doc/groups.xml:28-35
gap> Comm( [ (1,2), (2,3) ] );
(1,2,3)
gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] );
(1,5,6)
gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6));  ## the same
(1,5,6)

# doc/groups.xml:47-54
gap> D12 := DihedralGroup( 12 );
<pc group of size 12 with 3 generators>
gap> SetName( D12, "D12" ); 
gap> a := D12.1;;  b := D12.2;;  
gap> IsCommuting( a, b );
false

# doc/groups.xml:67-76
gap> ListOfPowers( 2, 20 );
[ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,
 32768, 65536, 131072, 262144, 524288, 1048576 ]
gap> ListOfPowers( (1,2,3)(4,5), 12 );
[ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (),
 (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ]
gap> ListOfPowers( D12.2, 6 );
[ f2, f3, f2*f3, f3^2, f2*f3^2, <identity> of ... ]

# doc/groups.xml:89-94
gap> GeneratorsAndInverses( D12 );
[ f1, f2, f3, f1, f2*f3^2, f3^2 ]
gap> GeneratorsAndInverses( SymmetricGroup(5) );     
[ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ]

# doc/groups.xml:114-132
gap> UpperFittingSeries( D12 );;
gap> List( last, StructureDescription );
[ "1", "C6", "D12" ]
gap> LowerFittingSeries( D12 );;
gap> List( last, StructureDescription );
[ "D12", "C3", "1" ]
gap> FittingLength( D12 );
2
gap> S4 := SymmetricGroup( 4 );;
gap> UpperFittingSeries( S4 );;
gap> List( last, StructureDescription );
[ "1", "C2 x C2", "A4", "S4" ]
gap> LowerFittingSeries( S4 );;
gap> List( last, StructureDescription );
[ "S4", "A4", "C2 x C2", "1" ]
gap> FittingLength( S4);
3

# doc/groups.xml:167-188
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

# doc/groups.xml:199-204
gap> Inverse( rc[3] ) = lc[3];
true
gap> Inverse( lc[2] ) = rc[2];
true

# doc/groups.xml:225-244
gap> G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );;
gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G );
[ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ]
gap> PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) );
d*c*b*a
gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;;
gap> d*c*b*a;
(1,2,3,4,5,6,7,8,9)
gap> ## note that it is easy to produce nonsense: 
gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) );
Warning: calling GroupHomomorphismByImagesNC without checks
[ (1,2,3) ] -> [ (8,9) ]
gap> IsGroupHomomorphism( epi );
true
gap> Image( epi, (1,2,3) );                                            
()
gap> Image( epi, (1,3,2) );
(8,9)

# doc/groups.xml:278-303
gap> s4 := Group( (1,2),(2,3),(3,4) );;
gap> s3 := Group( (5,6),(6,7) );;
gap> c3 := Subgroup( s3, [ (5,6,7) ] );;
gap> f := GroupHomomorphismByImages( s4, s3, 
>             [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );; 
gap> i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );; 
gap> Pfi := Pullback( f, i );
Group([ (2,3,4)(5,7,6), (1,2)(3,4) ])
gap> StructureDescription( Pfi );
"A4"
gap> info := PullbackInfo( Pfi );
rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]), 
  projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ], 
      [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] )
gap> g := (1,2,3)(5,6,7);; 
gap> ImageElm( info!.projections[1], g );
(1,2,3)
gap> ImageElm( info!.projections[2], g );
(5,6,7) 
gap> dp := info!.directProduct;; 
gap> a := ImageElm( Embedding( dp, 1 ), (1,4,3) );; 
gap> b := ImageElm( Embedding( dp, 2 ), (5,7,6) );; 
gap> a*b in Pfi;
true

# doc/groups.xml:347-369
gap> g1 := DihedralGroup( 8 );
<pc group of size 8 with 3 generators>
gap> c1 := Centre( g1 );
Group([ f3 ])
gap> cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) );
Group([ f1, f2, f5, f3, f4, f5 ])
gap> IsomorphismGroups( cp1, ExtraspecialGroup( 2^5, "+" ) ) <> fail;
true
gap> g2 := QuaternionGroup( 8 );
<pc group of size 8 with 3 generators>
gap> c2 := Centre( g2 );
Group([ y2 ])
gap> cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) );
Group([ f1, f2, f5, f3, f4, f5 ])
gap> IsomorphismGroups( cp1, ExtraspecialGroup( 2^5, "+" ) ) <> fail;
true
gap> info2 := CentralProductInfo( cp2 );
rec( phi := IdentityMapping( Group([ y2 ]) ), 
  projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] )
gap> Source( Embedding( Source( info2.projection ), 1 ) ) = g2;
true

# doc/groups.xml:402-423
gap> gens := [ (1,2,3,4), (1,2)(3,4) ];; 
gap> d8 := Group( gens );;
gap> SetName( d8, "d8" );
gap> c2 := Subgroup( d8, [ (2,4) ] );;
gap> SortedList( IdempotentEndomorphismsWithImage( gens, c2 ) );
[ [ (), (2,4) ], [ (2,4), () ] ]
gap> data:= IdempotentEndomorphismsData( d8 );;
gap> data.images[1];
[ [ (), () ] ]
gap> List( data.images, Length );
[ 1, 2, 2, 2, 2, 1 ]
gap> all:= IdempotentEndomorphisms( d8 );;
gap> Length( all );
10
gap> all[1];
[ (1,2,3,4), (1,2)(3,4) ] -> [ (), () ]
gap> Size( Image( all[1] ) );
1
gap> Last( all ) = IdentityMapping( d8 );
true

# doc/groups.xml:431-435
gap> q8 := QuaternionGroup( 8 );;
gap> IdempotentEndomorphisms( q8 );
[ [ x, y ] -> [ <identity> of ..., <identity> of ... ], [ x, y ] -> [ x, y ] ]

# doc/groups.xml:445-460
gap> c4 := Group( (1,2,3,4) );; 
gap> c2 := Group( (5,6) );; 
gap> f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );;
gap> c3 := Group( (1,2,3) );; 
gap> c6 := Group( (1,2,3,4,5,6) );; 
gap> f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );; 
gap> c4c3 := DirectProduct( c4, c3 ); 
Group([ (1,2,3,4), (5,6,7) ])
gap> c2c6 := DirectProduct( c2, c6 ); 
Group([ (1,2), (3,4,5,6,7,8) ])
gap> f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 ); 
[ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ]
gap> ImageElm( f, (1,4,3,2)(5,7,6) ); 
(1,2)(3,7,5)(4,8,6)

# doc/groups.xml:472-489
gap> c9 := Group( (1,2,3,4,5,6,7,8,9) );; 
gap> ac9 := AutomorphismGroup( c9 );; 
gap> q8 := QuaternionGroup( IsPermGroup, 8 );;
gap> aq8 := AutomorphismGroup( q8 );;
gap> A := DirectProductOfAutomorphismGroups( ac9, aq8 );
<group with 5 generators>
gap> genA := GeneratorsOfGroup( A );;
gap> G := Source( genA[1] );
Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)
(14,15,16,17) ])
gap> a := genA[1]*genA[5];  
[ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17) 
 ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17), 
  (10,11,12,13)(14,15,16,17) ]
gap> ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) );
(1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17)

#
gap> STOP_TEST("utils05.tst", 1);
