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gap> HeLP_Solver("4ti2");;
#I '4ti2' will be used from now on.
gap> OldHeLPInfoLevel := InfoLevel(HeLP_Info);;
gap> SetInfoLevel(HeLP_Info, 1);
gap> G := AlternatingGroup(5);;
gap> HeLP_ZC(CyclicGroup(6));
#I Since the given group is nilpotent the Zassenhaus Conjecture holds by a result of Al Weiss.
true
gap> HeLP_ZC(G);
true
gap> C := CharacterTable("A5");;
gap> HeLP_ZC(C);
true
gap> List(HeLP_sol, x -> Set(x));
[ [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ] ] ],,
[ [ [ 0, 1 ] ], [ [ 1, 0 ] ] ], [ ],,,, [ ],,,,, [ ],,,,,,,,,,,,,,, [ ]
]
gap> C := CharacterTable( "A6" );;
gap> HeLP_ZC(C);
#I (ZC) can't be solved, using the given data, for the orders: [ 6 ].
false
gap> Set(HeLP_sol[6]);
[ [ [ 1 ], [ 0, 1 ], [ -2, 2, 1 ] ], [ [ 1 ], [ 1, 0 ], [ -2, 1, 2 ] ] ]
gap> HeLP_sol[12];
[ ]
gap> G := SmallGroup(48,30);;
gap> HeLP_ZC(G);
#I (ZC) can't be solved, using the given data, for the orders: [ 4 ].
false
gap> Size(HeLP_sol[4]);
10
gap> C1 := CharacterTable(PSL(2,7));;
gap> HeLP_PQ(C1);
#I The Brauer tables for the following primes are not available: [ 2, 3, 7 ].
#I (PQ) can't be solved, using the given data, for the orders: [ 6 ].
false
gap> C2 := CharacterTable("L2(7)");;
gap> HeLP_PQ(C2);
true
gap> C := CharacterTable("L2(49).2_1");;
gap> HeLP_WithGivenOrder(Irr(C), 7);;
#I Number of solutions for elements of order 7: 1; stored in HeLP_sol[7].
gap> HeLP_WithGivenOrder(Irr(C){[2]}, 14);
#I The given data admit infinitely many solutions for elements of order 14.
"infinite"
gap> HeLP_WithGivenOrder(Irr(C){[2,44]}, 14);
#I Number of solutions for elements of order 14: 0; stored in HeLP_sol[14].
[ ]
gap> C := CharacterTable("A5");;
gap> chi := Irr(C)[2];; psi := Irr(C)[4];;
gap> HeLP_WithGivenOrderAndPAAndSpecificSystem([[chi, 1], [chi, 2]], 5, [ ], true); # Ist das eindeutig? Oder muss ein Set rein?
[ [ [ [ 0, 1 ] ], [ [ 1, 0 ] ] ], [ [ -3/5, 2/5 ], [ 2/5, -3/5 ] ], [ 3/5, 3/5 ] ]
gap> sol5 := HeLP_WithGivenOrderAndPAAndSpecificSystem([[chi, 1], [chi, 2]], 5, [ ]); # Hier eindeutig?
[ [ [ 0, 1 ] ], [ [ 1, 0 ] ] ]
gap> HeLP_WithGivenOrderAndPAAndSpecificSystem([psi], 2*5, [[1], sol5[1][1]], true);
[ [ ], [ [ 0, -2/5, -2/5 ], [ 0, -1/10, -1/10 ], [ 0, 1/10, 1/10 ],
[ 0, -1/10, -1/10 ], [ 0, 1/10, 1/10 ], [ 0, 2/5, 2/5 ],
[ 0, 1/10, 1/10 ], [ 0, -1/10, -1/10 ], [ 0, 1/10, 1/10 ],
[ 0, -1/10, -1/10 ] ], [ 0, 1/2, 1/2, 1/2, 1/2, 0, 1/2, 1/2, 1/2, 1/2 ] ]
gap> HeLP_WithGivenOrderAndPAAndSpecificSystem([[psi, 0], [psi, 2], [psi, 5]], 2*5, [[1], sol5[2][1]], true);
[ [ ], [ [ 0, -2/5, -2/5 ], [ 0, 1/10, 1/10 ], [ 0, 2/5, 2/5 ] ], [ 0, 1/2, 0 ] ]
gap> C := CharacterTable("A6");;
gap> Set(HeLP_WithGivenOrder(C, 6));
#I Number of solutions for elements of order 6: 2; stored in HeLP_sol[6].
[ [ [ 1 ], [ 0, 1 ], [ -2, 2, 1 ] ], [ [ 1 ], [ 1, 0 ], [ -2, 1, 2 ] ] ]
gap> Set(HeLP_WithGivenOrderSConstant(C, 2, 3));
#I Number of solutions for elements of order 6: 2.
[ [ [ 0, 1 ], [ -2, 2, 1 ] ], [ [ 1, 0 ], [ -2, 1, 2 ] ] ]
gap> HeLP_WithGivenOrderSConstant(C, 3, 2);
#I Number of solutions for elements of order 6: 1.
[ [ [ 1 ], [ 3, -2 ] ] ]
gap> HeLP_WithGivenOrderAndPA(C, 6, [[1],[1,0]]);
#I Number of solutions for elements of order 6 with these partial augmentations for the powers: 1.
[ [ [ 1 ], [ 1, 0 ], [ -2, 1, 2 ] ] ]
gap> C := CharacterTable("Sz(32)");;
gap> HeLP_WithGivenOrderSConstant(C mod 2, 31, 5);
#I Number of solutions for elements of order 155: 0; stored in HeLP_sol[155].
[ ]
gap> IsBound(HeLP_sol[31]);
false
gap> C := CharacterTable("L2(7)");;
gap> HeLP_WithGivenOrder(C,6);
#I Number of solutions for elements of order 6: 1; stored in HeLP_sol[6].
[ [ [ 1 ], [ 1 ], [ -2, 3 ] ] ]
gap> HeLP_AllOrders(C);
true
gap> C := CharacterTable("A12");;
gap> HeLP_WithGivenOrder(Irr(C){[2, 4, 7]}, 2);;
#I Number of solutions for elements of order 2: 37; stored in HeLP_sol[2].
gap> HeLP_WithGivenOrderSConstant(C mod 3,11,2);
#I Number of solutions for elements of order 22: 0; stored in HeLP_sol[22].
[ ]
gap> HeLP_WithGivenOrder(Irr(C mod 2){[2, 3, 4, 6]}, 3);;
#I Number of solutions for elements of order 3: 99; stored in HeLP_sol[3].
gap> HeLP_WithGivenOrderSConstant(C mod 2, 11, 3);
#I Number of solutions for elements of order 33: 0; stored in HeLP_sol[33].
[ ]
gap> HeLP_AllOrdersPQ(C);
true
gap> C := CharacterTable("M11");;
gap> HeLP_WithGivenOrder(C,8);;
#I Number of solutions for elements of order 8: 36; stored in HeLP_sol[8].
gap> HeLP_sol[8] := HeLP_WagnerTest(8);;
gap> Size(HeLP_sol[8]);
24
gap> HeLP_WithGivenOrder(C,12);;
#I Number of solutions for elements of order 12: 7; stored in HeLP_sol[12].
gap> HeLP_sol[12] := HeLP_WagnerTest(12);;
gap> HeLP_sol[4] := HeLP_WagnerTest(4);;
gap> HeLP_WithGivenOrder(C,12);;
#I Number of solutions for elements of order 12: 3; stored in HeLP_sol[12].
gap> HeLP_sol[12] := HeLP_WagnerTest(12);
[ ]
gap> C := CharacterTable("M22");;
gap> HeLP_WagnerTest(12, [ [ [1], [1], [1,0], [0,0,1], [-3,3,2,3,-4] ] ],C);
[ ]
gap> G := SmallGroup(96,187);;
gap> C := CharacterTable(G);;
gap> HeLP_WithGivenOrder(C,4);;
#I Number of solutions for elements of order 4: 34; stored in HeLP_sol[4].
gap> Size(HeLP_WagnerTest(4));
4
gap> C := CharacterTable("A6");;
gap> HeLP_WithGivenOrder(C, 4);;
#I Number of solutions for elements of order 4: 4; stored in HeLP_sol[4].
gap> Size(HeLP_VerifySolution(C mod 3, 4));
#I Number of solutions for elements of order 4: 2; stored in HeLP_sol[4].
2
gap> Size(HeLP_sol[4]);
2
gap> C := CharacterTable(SmallGroup(160,91));;
gap> HeLP_WithGivenOrder(C,4);;
#I Number of solutions for elements of order 4: 22; stored in HeLP_sol[4].
gap> HeLP_WithGivenOrder(C,10);;
#I Number of solutions for elements of order 10: 6; stored in HeLP_sol[10].
gap> LP := HeLP_PossiblePartialAugmentationsOfPowers(20);;
gap> Size(LP);
44
gap> C := CharacterTable("A6");;
gap> HeLP_WithGivenOrder(C, 6);;
#I Number of solutions for elements of order 6: 2; stored in HeLP_sol[6].
gap> HeLP_sol[6] := Set(HeLP_sol[6]);;
gap> chi := Irr(C)[2];;
gap> HeLP_MultiplicitiesOfEigenvalues(chi, 6, Set(HeLP_sol[6])[2]);
[ 1, 0, 1, 2, 1, 0 ]
gap> HeLP_CharacterValue(chi, 6, Set(HeLP_sol[6])[2][3]);
-2
gap> HeLP_CharacterValue(chi, 6, [-2,1,2]);
-2
gap> HeLP_CharacterValue(chi, 6, [-2,2,1]);
1
gap> HeLP_Reset();
gap> HeLP_sol;
[ [ [ [ 1 ] ] ] ]
gap> SetInfoLevel(HeLP_Info, OldHeLPInfoLevel);
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