411 lines
10 KiB
Scilab
411 lines
10 KiB
Scilab
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#############################################################################
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##
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#W core_methods.tst Testing core methods G. P. Nagy / P. Vojtechovsky
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##
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#H @(#)$Id: core_methods.tst, v 3.3.0 2016/10/26 gap Exp $
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##
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#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
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#Y P. Vojtechovsky (University of Denver, USA)
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##
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gap> START_TEST("LOOPS, core_methods: testing core methods");
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# TESTING VIEW AND PRINT MODE, AND LATIN SQUARE FUNCTIONS
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gap> T := [ [ 2, 1 ], [ 1, 2 ] ];;
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gap> IsQuasigroupTable( T );
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true
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gap> IsLoopTable( T );
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false
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gap> Q := QuasigroupByCayleyTable( T );
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<quasigroup of order 2>
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gap> Elements( Q );
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[ q1, q2 ]
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gap> L := IntoLoop( Q );
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<loop of order 2>
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gap> Elements( L );
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[ l1, l2 ]
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gap> L.1;
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l1
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# TESTING MORE CONVERSION FUNCTIONS
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gap> G := IntoGroup( Q );
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Group([ (), (1,2) ])
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gap> G := IntoGroup( L );
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Group([ (), (1,2) ])
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gap> IntoQuasigroup( G );
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<quasigroup of order 2>
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gap> IntoLoop( G );
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<loop of order 2>
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gap> IntoLoop( Group( (1,2,3), (1,2), (1,4) ) );
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<loop of order 24>
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gap> PrincipalLoopIsotope( Q, Elements(Q)[1], Elements(Q)[1] );
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<loop of order 2>
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gap> CanonicalCopy( QuasigroupByCayleyTable( [[2,3],[3,2]] ) );
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<quasigroup of order 2>
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# TESTING DIRECT PRODUCTS AND OPPOSITES
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gap> L := MoufangLoop( 12, 1 );;
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gap> DirectProduct( L );
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<Moufang loop 12/1>
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gap> DirectProduct( L, L, L );
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<loop of order 1728>
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gap> DirectProduct( L, Group( (1,2,3) ) );
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<loop of order 36>
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gap> DirectProduct( Group( (1,2,3) ), L, Group( (1,2,3,4) ) );
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<loop of order 144>
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gap> Q := QuasigroupByCayleyTable([[2,1],[1,2]]);;
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gap> DirectProduct( Q, Q );
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<quasigroup of order 4>
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gap> DirectProduct( Q, L, Group((1,2)) );
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<quasigroup of order 48>
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gap> OppositeLoop( L );
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<loop of order 12>
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# TESTING BASIC ATTRIBUTES
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gap> One( L );
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l1
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gap> Size( L );
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12
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gap> CayleyTable( Q );
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[ [ 2, 1 ], [ 1, 2 ] ]
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gap> Exponent( L );
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6
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gap> Opposite( L );
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<loop of order 12>
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# TESTING BASIC ARITHMETIC OPERATIONS
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gap> eL := Elements( L );;
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gap> eL[ 2 ]*eL[ 3 ]*eL[ 7 ] = (eL[ 2 ]*eL[ 3 ])* eL[ 7 ];
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true
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gap> eL[ 2 ]*eL[ 3 ]*eL[ 7 ] = eL[ 2 ]*(eL[ 3 ]* eL[ 7 ]);
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false
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gap> LeftDivision( eL[ 2 ], eL[ 3 ] );
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l4
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gap> RightDivision( eL[ 2 ], eL[ 3 ] );
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l6
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gap> LeftInverse( eL[ 2 ] ) = RightInverse( eL[ 2 ] );
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true
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gap> Associator( eL[ 2 ], eL[ 3 ], eL[ 4 ] );
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l1
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gap> Commutator( eL[ 2 ], eL[ 3 ] );
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l5
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# TESTING GENERATORS
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gap> GeneratorsOfLoop( L );
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[ l1, l2, l3, l4, l5, l6, l7, l8, l9, l10, l11, l12 ]
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gap> GeneratorsSmallest( L );
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[ l10, l11, l12 ]
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gap> SmallGeneratingSet( L );
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[ l2, l3, l7 ]
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# TESTING SECTIONS AND TRANSLATIONS
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gap> LeftSection( L );
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[ (), (1,2)(3,4)(5,6)(7,8)(9,12)(10,11), (1,3,5)(2,6,4)(7,9,11)(8,10,12),
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(1,4)(2,5)(3,6)(7,10)(8,9)(11,12), (1,5,3)(2,4,6)(7,11,9)(8,12,10),
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(1,6)(2,3)(4,5)(7,12)(8,11)(9,10), (1,7)(2,8)(3,11)(4,10)(5,9)(6,12),
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(1,8)(2,7)(3,12)(4,9)(5,10)(6,11), (1,9)(2,12)(3,7)(4,8)(5,11)(6,10),
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(1,10)(2,11)(3,8)(4,7)(5,12)(6,9), (1,11)(2,10)(3,9)(4,12)(5,7)(6,8),
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(1,12)(2,9)(3,10)(4,11)(5,8)(6,7) ]
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gap> RightSection( L );
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[ (), (1,2)(3,6)(4,5)(7,8)(9,12)(10,11), (1,3,5)(2,4,6)(7,11,9)(8,12,10),
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(1,4)(2,3)(5,6)(7,10)(8,9)(11,12), (1,5,3)(2,6,4)(7,9,11)(8,10,12),
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(1,6)(2,5)(3,4)(7,12)(8,11)(9,10), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12),
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(1,8)(2,7)(3,10)(4,9)(5,12)(6,11), (1,9)(2,12)(3,11)(4,8)(5,7)(6,10),
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(1,10)(2,11)(3,12)(4,7)(5,8)(6,9), (1,11)(2,10)(3,7)(4,12)(5,9)(6,8),
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(1,12)(2,9)(3,8)(4,11)(5,10)(6,7) ]
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gap> LeftTranslation( L, eL[ 3 ] );
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(1,3,5)(2,6,4)(7,9,11)(8,10,12)
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gap> RightTranslation( L, eL[ 3 ] );
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(1,3,5)(2,4,6)(7,11,9)(8,12,10)
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# TESTING MULTIPLICATION GROUPS AND INNER MAPPING GROUPS
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gap> LeftMultiplicationGroup( L );
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<permutation group with 12 generators>
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gap> RightMultiplicationGroup( L );
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<permutation group with 12 generators>
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gap> Size( MultiplicationGroup( L ) );
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2592
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gap> Size( InnerMappingGroup( L ) );
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216
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gap> MiddleInnerMappingGroup( L );
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<permutation group with 12 generators>
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# TESTING LOOP BY LEFT/RIGHT SECTION
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gap> LoopByRightSection( [ (), (1,2)(3,4,5), (1,3,5)(2,4), (1,4,3)(2,5), (1,5,4)(2,3) ] );
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<loop of order 5>
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gap> S := Subloop( MoufangLoop( 12, 1 ), [ 3 ] );;
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gap> LoopByLeftSection( LeftSection( S ) );
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<loop of order 3>
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gap> LoopByRightSection( RightSection( S ) );
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<loop of order 3>
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gap> QuasigroupByLeftSection( LeftSection( S ) );
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<quasigroup of order 3>
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gap> QuasigroupByRightSection( RightSection( S ) );
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<quasigroup of order 3>
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gap> CayleyTableByPerms( LeftSection( S ) );
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[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
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# TESTING LOOP BY RIGHT FOLDER
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gap> LOOPS_Shift := function( p )
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> local ls;
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> ls := ListPerm( p );
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> ls := Concatenation( [1,2,3,4,5], List( ls, x -> x + 5 ) );
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> return PermList( ls );
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> end;
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function( p ) ... end
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gap> A := AlternatingGroup( 5 );
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Alt( [ 1 .. 5 ] )
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gap> G := DirectProduct( A, A );
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Group([ (1,2,3,4,5), (3,4,5), (6,7,8,9,10), (8,9,10) ])
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gap> H := Subgroup( G, [ (1,2,3), (2,3,4), (6,7,8,9,10) ] );
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Group([ (1,2,3), (2,3,4), (6,7,8,9,10) ])
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gap> T := List( A, x -> x * LOOPS_Shift(x)^(-1) );;
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gap> LoopByRightFolder( G, H, T );
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<loop of order 60>
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gap> QuasigroupByRightFolder( G, H, T );
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<quasigroup of order 60>
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# TESTING RANDOM QUASIGROUPS AND LOOPS
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gap> RandomQuasigroup( 10 );
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<quasigroup of order 10>
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gap> RandomQuasigroup( 10, 100 );
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<quasigroup of order 10>
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gap> RandomLoop( 11 );
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<loop of order 11>
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gap> RandomLoop( 30, 1000 );
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<loop of order 30>
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gap> RandomNilpotentLoop( [2, 3, 5] );
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<loop of order 30>
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gap> RandomNilpotentLoop( [2, CyclicGroup(3), 6] );
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<loop of order 36>
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# TESTING SUBQUASIGROUPS AND SUBLOOPS
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gap> L := MoufangLoop( 32, 5 );;
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gap> Length( AllSubloops( L ) );
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90
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gap> S := Subloop( L, [ Elements( L )[ 2 ], Elements( L )[ 25 ] ] );
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<loop of order 8>
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gap> L = Parent( S );
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true
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gap> L = Parent( L );
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true
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gap> PosInParent( S );
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[ 1, 2, 5, 8, 22, 25, 29, 31 ]
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gap> IsSubloop( L, S );
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true
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gap> CayleyTable( S );
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[ [ 1, 2, 5, 8, 22, 25, 29, 31 ], [ 2, 5, 8, 1, 31, 22, 25, 29 ],
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[ 5, 8, 1, 2, 29, 31, 22, 25 ], [ 8, 1, 2, 5, 25, 29, 31, 22 ],
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[ 22, 25, 29, 31, 1, 2, 5, 8 ], [ 25, 29, 31, 22, 8, 1, 2, 5 ],
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[ 29, 31, 22, 25, 5, 8, 1, 2 ], [ 31, 22, 25, 29, 2, 5, 8, 1 ] ]
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gap> LeftTranslation( S, Elements( S )[ 2 ] );
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(1,2,5,8)(22,31,29,25)
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gap> SS := Subloop( S, [ Elements( S )[ 2 ] ] );
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<loop of order 4>
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gap> Parent( SS ) = L;
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true
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gap> CayleyTable( SS );
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[ [ 1, 2, 5, 8 ], [ 2, 5, 8, 1 ], [ 5, 8, 1, 2 ], [ 8, 1, 2, 5 ] ]
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gap> IsSubloop( S, SS );
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true
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gap> AllSubquasigroups( QuasigroupByCayleyTable( [[2,1],[1,2]] ) );
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[ <quasigroup of order 2>, <quasigroup of order 1> ]
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# TESTING NUCLEUS, COMMUTANT, CENTER
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gap> LeftNucleus( L ) = NucleusOfLoop( L );
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true
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gap> MiddleNucleus( L ) = RightNucleus( L );
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true
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gap> Commutant( L );
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[ l1, l4, l5, l11 ]
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gap> Center( L );
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<associative loop of order 4>
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gap> AssociatorSubloop( L );
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<loop of order 2>
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# TESTING COMMUTATIVITY AND GENERALIZATIONS
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gap> IsAssociative( L );
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false
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gap> IsCommutative( L );
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false
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gap> IsCommutative( Q );
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true
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gap> IsPowerAssociative( L );
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true
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gap> IsDiassociative( L );
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true
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# TESTING INVERSE PROPERTIES
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gap> B := LeftBolLoop( 8, 1 );;
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gap> HasLeftInverseProperty( B );
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true
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gap> HasRightInverseProperty( B );
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false
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gap> HasInverseProperty( B );
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false
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gap> HasTwosidedInverses( B );
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true
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gap> HasAutomorphicInverseProperty( B );
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true
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gap> HasAntiautomorphicInverseProperty( B );
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false
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# TESTING PROPERTIES OF QUASIGROUPS
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gap> IsSemisymmetric( Q );
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true
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gap> IsTotallySymmetric( Q );
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true
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gap> IsTotallySymmetric( B );
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false
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gap> IsIdempotent( Q );
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false
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gap> IsSteinerQuasigroup( Q );
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false
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gap> IsUnipotent( Q );
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true
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gap> IsLeftDistributive( B );
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false
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gap> IsRightDistributive( B );
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false
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gap> IsDistributive( B );
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false
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gap> IsEntropic( Q );
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true
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gap> IsMedial( Q );
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true
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# TESTING LOOPS OF BOL-MOUFANG TYPE
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gap> L := DirectProduct( MoufangLoop( 12, 1 ), Group( (1,2)(3,4), (1,3)(2,4) ) );
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<loop of order 48>
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gap> IsLeftAlternative( L );
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true
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gap> IsRightAlternative( L );
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true
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gap> L;
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<alternative loop of order 48>
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gap> IsLeftNuclearSquareLoop( L );
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false
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gap> IsRightNuclearSquareLoop( L );
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false
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gap> IsMiddleNuclearSquareLoop( L );
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false
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gap> IsNuclearSquareLoop( L );
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false
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gap> IsLCLoop( L );
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false
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gap> IsRCLoop( L );
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false
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gap> IsCLoop( L );
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false
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gap> IsFlexible( L );
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true
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gap> IsLeftBolLoop( L );
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true
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gap> L;
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<left Bol loop of order 48>
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gap> IsRightBolLoop( L );
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true
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gap> L;
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<Moufang loop of order 48>
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gap> IsExtraLoop( L );
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false
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# TESTING CONJUGACY CLOSED LOOPS
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gap> IsLCCLoop( L ); IsLeftConjugacyClosedLoop( L );
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false
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false
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gap> IsRCCLoop( L ); IsRightConjugacyClosedLoop( L );
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false
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false
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gap> IsCCLoop( L ); IsConjugacyClosedLoop( L );
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false
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false
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||
|
|
|
||
|
|
# TESTING BRUCK AND STEINER LOOPS
|
||
|
|
|
||
|
|
gap> IsLeftBruckLoop( B );
|
||
|
|
true
|
||
|
|
gap> IsRightBruckLoop( B );
|
||
|
|
false
|
||
|
|
gap> IsLeftKLoop( B );
|
||
|
|
true
|
||
|
|
gap> IsRightKLoop( B );
|
||
|
|
false
|
||
|
|
gap> IsSteinerLoop( B );
|
||
|
|
false
|
||
|
|
|
||
|
|
# TESTING A-LOOPS
|
||
|
|
|
||
|
|
gap> IsLeftALoop( B );
|
||
|
|
true
|
||
|
|
gap> IsRightALoop( B );
|
||
|
|
true
|
||
|
|
gap> IsMiddleALoop( B );
|
||
|
|
false
|
||
|
|
gap> IsALoop( B );
|
||
|
|
false
|
||
|
|
|
||
|
|
# TESTING NORMALITY
|
||
|
|
|
||
|
|
gap> L := MoufangLoop( 32, 27 );;
|
||
|
|
gap> S := Subloop( L, [ L.3, L.4 ] );;
|
||
|
|
gap> IsNormal( L, S );
|
||
|
|
true
|
||
|
|
gap> FactorLoop( L, S );
|
||
|
|
<loop of order 4>
|
||
|
|
gap> NaturalHomomorphismByNormalSubloop( L, S );
|
||
|
|
MappingByFunction( <Moufang loop 32/27>, <loop of order
|
||
|
|
4>, function( x ) ... end )
|
||
|
|
gap> S := Subloop( L, [ Elements( L )[ 7 ] ] );;
|
||
|
|
gap> IsNormal( L, S );
|
||
|
|
false
|
||
|
|
gap> NormalClosure( L, S );
|
||
|
|
<loop of order 8>
|
||
|
|
|
||
|
|
# TESTING NILPOTENCY (MORE TESTING IN FILE nilpot.tst)
|
||
|
|
|
||
|
|
gap> IsNilpotent( L );
|
||
|
|
true
|
||
|
|
gap> IsStronglyNilpotent( L );
|
||
|
|
true
|
||
|
|
gap> UpperCentralSeries( L );
|
||
|
|
[ <loop of order 32>, <loop of order 4>, <loop of order 2>,
|
||
|
|
<associative loop of order 1> ]
|
||
|
|
gap> LowerCentralSeries( L );
|
||
|
|
[ <Moufang loop 32/27>, <loop of order 4>, <loop of order 2>,
|
||
|
|
<associative loop of order 1> ]
|
||
|
|
gap> NilpotencyClassOfLoop( L );
|
||
|
|
3
|
||
|
|
|
||
|
|
# TESTING SOLVABILITY
|
||
|
|
|
||
|
|
gap> IsSolvable( L );
|
||
|
|
true
|
||
|
|
gap> DerivedSubloop( L );
|
||
|
|
<loop of order 4>
|
||
|
|
gap> DerivedLength( L );
|
||
|
|
2
|
||
|
|
gap> FrattiniSubloop( L );
|
||
|
|
<loop of order 4>
|
||
|
|
gap> FrattinifactorSize( L );
|
||
|
|
8
|
||
|
|
|
||
|
|
gap> STOP_TEST( "core_methods.tst", 10000000 );
|