67 lines
1.8 KiB
Scilab
67 lines
1.8 KiB
Scilab
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#############################################################################
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##
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#W nilpot.tst Testing nilpotency G. P. Nagy / P. Vojtechovsky
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##
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#H @(#)$Id: nilpot.tst, v 3.2.0 2015/11/22 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, nilpot: nilpotency and triality group");
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# GENERAL NILPOTENCY
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gap> L := LoopByCayleyTable(
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> [ [ 1, 2, 3, 4, 5, 6, 7, 8 ], [ 2, 1, 4, 3, 6, 5, 8, 7 ],
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> [ 3, 4, 1, 2, 7, 8, 5, 6 ], [ 4, 6, 2, 8, 1, 7, 3, 5 ],
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> [ 5, 3, 7, 1, 8, 2, 6, 4 ], [ 6, 5, 8, 7, 2, 1, 4, 3 ],
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> [ 7, 8, 5, 6, 3, 4, 1, 2 ], [ 8, 7, 6, 5, 4, 3, 2, 1 ] ] );
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<loop of order 8>
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gap> Center(L);
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<associative loop of order 2>
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gap> LeftNucleus(L);
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<associative loop of order 2>
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gap> RightNucleus(L);
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<associative loop of order 4>
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gap> IsNilpotent(L);
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true
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gap> NilpotencyClassOfLoop(L);
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2
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gap> IsomorphismLoops(L,LeftBolLoop(8,2));
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(3,8,4,6,5,7)
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# NILPOTENCY FOR MOUFANG LOOPS
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gap> L:=MoufangLoop(24,1);
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<Moufang loop 24/1>
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gap> Center(L);
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<associative loop of order 2>
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gap> IsNilpotent(L);
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false
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gap> NilpotencyClassOfLoop(L);
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fail
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gap> L:=MoufangLoop(32,32);
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<Moufang loop 32/32>
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gap> Center(L);
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<associative loop of order 2>
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# TRIALITY GROUPS
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gap> tr:=TrialityPermGroup(L);;
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gap> [ 33^tr.rho, 33^tr.sigma ];
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[ 65, 1 ]
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gap> Size(Center(tr.group));
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4
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gap> tr_pc:=TrialityPcGroup(L);
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rec( group := <Triality pc group of order 2^15>, rho := f2, sigma := f1 )
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gap> Size(Centralizer(tr.group,tr.sigma));
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1024
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gap> Size(Centralizer(tr.group,tr.rho));
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32
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gap> TrialityPermGroup(PSL(2,5));;
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gap> Size(last.group);
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216000
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gap> STOP_TEST( "nilpot.tst", 10000000 );
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