gpnobjmap.tst

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#W  gpnobjmap.tst                 XMOD test file                Chris Wensley
##                                                               Alper Odabas
#Y  Copyright (C) 2001-2017, Chris Wensley et al, 
#Y  School of Computer Science, Bangor University, U.K. 
##
gap> START_TEST( "XMod package: gpnobjmap.tst" );
gap> saved_infolevel_xmod := InfoLevel( InfoXMod );; 
gap> SetInfoLevel( InfoXMod, 0 );;
gap> saved_infolevel_groupoids := InfoLevel( InfoGroupoids );; 
gap> SetInfoLevel( InfoGroupoids, 0 );;

gap> s3 := SmallGroup( 6, 1 );;
gap> homs := AllHomomorphisms( s3, s3 );;
gap> idem := Filtered( homs, i -> CompositionMapping(i,i) = i );
[ [ f1, f2 ] -> [ <identity> of ..., <identity> of ... ],
  [ f1, f2 ] -> [ f1, <identity> of ... ],
  [ f1, f2 ] -> [ f1*f2^2, <identity> of ... ],
  [ f1, f2 ] -> [ f1*f2, <identity> of ... ], [ f1, f2 ] -> [ f1, f2 ] ]
gap> pc1 := PreCat1GroupByEndomorphisms( idem[1], idem[1] );
[Group( [ f1, f2 ] )=>Group( [ <identity> of ..., <identity> of ... ] )]
gap> pc2 := PreCat1GroupByEndomorphisms( idem[2], idem[2] );
[Group( [ f1, f2 ] )=>Group( [ f1, <identity> of ... ] )]
gap> CC12 := CatnGroup( [ pc1, pc2 ] );
generating (pre-)cat1-groups:
1 : [Group( [ f1, f2 ] )=>Group( [ <identity> of ..., <identity> of ... ] )]
2 : [Group( [ f1, f2 ] )=>Group( [ f1, <identity> of ... ] )]
gap> HigherDimension( CC12 );
3

gap> pc3 := PreCat1GroupByEndomorphisms( idem[5], idem[5] );
[Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
gap> CC233 := CatnGroup( [pc2, pc3, pc3] );
generating (pre-)cat1-groups:
1 : [Group( [ f1, f2 ] )=>Group( [ f1, <identity> of ... ] )]
2 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
3 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]

gap> IsPreCatnGroup( CC233 );
true
gap> IsCatnGroup( CC233 );
true

gap> CC5 := CatnGroup( [ Cat1Group(8,2,1), Cat1Group(8,2,2),
>          Cat1Group(8,2,4), Cat1Group(8,2,6), Cat1Group(8,2,6) ] );
generating (pre-)cat1-groups:
1 : [C4 x C2=>Group( [ <identity> of ..., <identity> of ..., <identity> of ... ] )]
2 : [C4 x C2=>Group( [ <identity> of ..., f2 ] )]
3 : [C4 x C2=>Group( [ f1, <identity> of ... ] )]
4 : [C4 x C2=>C4 x C2]
5 : [C4 x C2=>C4 x C2]

gap> Display( CC5 );
generating (pre-)cat1-groups:
1 :
Cat1-group :-
: Source group C4 x C2 has generators:
  [ f1, f2, f3 ]
: Range group has generators:
  [ <identity> of ..., <identity> of ..., <identity> of ... ]
: tail homomorphism maps source generators to:
  [ <identity> of ..., <identity> of ..., <identity> of ... ]
: head homomorphism maps source generators to:
  [ <identity> of ..., <identity> of ..., <identity> of ... ]
: range embedding maps range generators to:
  [ <identity> of ..., <identity> of ..., <identity> of ... ]
: kernel has generators:
  [ f1, f2, f3 ]
: boundary homomorphism maps generators of kernel to:
  [ <identity> of ..., <identity> of ..., <identity> of ... ]
: kernel embedding maps generators of kernel to:
  [ f1, f2, f3 ]
: associated crossed module is [Group( [ f1, f2, f3 ] )->Group(
[ <identity> of ..., <identity> of ..., <identity> of ... ] )]

2 :
Cat1-group :-
: Source group C4 x C2 has generators:
  [ f1, f2, f3 ]
: Range group has generators:
  [ <identity> of ..., f2 ]
: tail homomorphism maps source generators to:
  [ <identity> of ..., f2, <identity> of ... ]
: head homomorphism maps source generators to:
  [ <identity> of ..., f2, <identity> of ... ]
: range embedding maps range generators to:
  [ <identity> of ..., f2 ]
: kernel has generators:
  [ f1, f3 ]
: boundary homomorphism maps generators of kernel to:
  [ <identity> of ..., <identity> of ... ]
: kernel embedding maps generators of kernel to:
  [ f1, f3 ]
: associated crossed module is [Group( [ f1, f3 ] )->Group(
[ <identity> of ..., f2 ] )]

3 :
Cat1-group :-
: Source group C4 x C2 has generators:
  [ f1, f2, f3 ]
: Range group has generators:
  [ f1, <identity> of ... ]
: tail homomorphism maps source generators to:
  [ f1, <identity> of ..., f3 ]
: head homomorphism maps source generators to:
  [ f1, <identity> of ..., f3 ]
: range embedding maps range generators to:
  [ f1, <identity> of ... ]
: kernel has generators:
  [ f2 ]
: boundary homomorphism maps generators of kernel to:
  [ <identity> of ... ]
: kernel embedding maps generators of kernel to:
  [ f2 ]
: associated crossed module is [Group( [ f2 ] )->Group(
[ f1, <identity> of ... ] )]

4 :
Cat1-group [C4 x C2=>C4 x C2] :-
: Source group C4 x C2 has generators:
  [ f1, f2, f3 ]
: Range group C4 x C2 has generators:
  [ f1, f2, f3 ]
: tail homomorphism maps source generators to:
  [ f1, f2, f3 ]
: head homomorphism maps source generators to:
  [ f1, f2, f3 ]
: range embedding maps range generators to:
  [ f1, f2, f3 ]
: the kernel is trivial.
: associated crossed module is [Group( <identity> of ... )->C4 x C2]

5 :
Cat1-group [C4 x C2=>C4 x C2] :-
: Source group C4 x C2 has generators:
  [ f1, f2, f3 ]
: Range group C4 x C2 has generators:
  [ f1, f2, f3 ]
: tail homomorphism maps source generators to:
  [ f1, f2, f3 ]
: head homomorphism maps source generators to:
  [ f1, f2, f3 ]
: range embedding maps range generators to:
  [ f1, f2, f3 ]
: the kernel is trivial.
: associated crossed module is [Group( <identity> of ... )->C4 x C2]

gap> CC6 := Cat2Group( Cat1Group(6,2,2), Cat1Group(6,2,3) );
generating (pre-)cat1-groups:
1 : [C6=>Group( [ f1 ] )]
2 : [C6=>Group( [ f2 ] )]

gap> IsCat2Group( CC6 );
true

gap> xsCC6 := CrossedSquareOfCat2Group( CC6 );
crossed square with:
      up = [Group( () )->Group( [ (1,2) ] )]
    left = [Group( () )->Group( [ (), (3,4,5) ] )]
    down = [Group( [ (), (3,4,5) ] )->Group( () )]
   right = [Group( [ (1,2) ] )->Group( () )]
   
gap> IsCrossedSquare( xsCC6 );
true

## CCconj := Cat2GroupOfCrossedSquare( XSconj );
## this command produces an error! 
## IsCat2Group( CCconj );
## true

gap> idCC233 := IdentityMapping( CC233 );
<mapping: generating (pre-)cat1-groups:
1 : [Group( [ f1, f2 ] )=>Group( [ f1, <identity> of ... ] )]
2 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
3 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
 -> generating (pre-)cat1-groups:
1 : [Group( [ f1, f2 ] )=>Group( [ f1, <identity> of ... ] )]
2 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
3 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
 >
 
gap> Display( idCC233 );
Morphism of pre-cat3-groups :- 
: Source has generating (pre-)cat1-groups:
1 : [Group( [ f1, f2 ] )=>Group( [ f1, <identity> of ... ] )]
2 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
3 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]

: Range has generating (pre-)cat1-groups:
1 : [Group( [ f1, f2 ] )=>Group( [ f1, <identity> of ... ] )]
2 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]
3 : [Group( [ f1, f2 ] )=>Group( [ f1, f2 ] )]

: MappingGeneratorsImages for the source homomorphisms:
1 : [ [ f1, f2 ], [ f1, f2 ] ]
2 : [ [ f1, f2 ], [ f1, f2 ] ]
3 : [ [ f1, f2 ], [ f1, f2 ] ]

gap> IsBijective( idCC233 );
true

gap> SetInfoLevel( InfoXMod, saved_infolevel_xmod );; 
gap> SetInfoLevel( InfoGroupoids, saved_infolevel_groupoids );; 
gap> STOP_TEST( "gpnobjmap.tst", 10000 );

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#E  gpnobjmap.tst . . . . . . . . . . . . . . . . . . . . . . . . . ends here