Section 1, continued
gap> g:= Group([(1,3), (1,4,5)]); Group([ (1,3), (1,4,5) ]) gap> KnownAttributesOfObject(g); [ "LargestMovedPoint", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement" ] gap> Size(g); 24 gap> KnownAttributesOfObject(g); [ "Size", "OneImmutable", "LargestMovedPoint", "NrMovedPoints", "MovedPoints", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement", "Pcgs", "GeneralizedPcgs", "StabChainMutable", "StabChainOptions" ]
Section 1 Appendix: Three Representations of the Group of Rotations of a Cube
gap> G8 := SymmetricGroup(8); Sym( [ 1 .. 8 ] ) gap> f8 := (1,2,3,4)(5,6,7,8); (1,2,3,4)(5,6,7,8) gap> l8 := (1,5,6,2)(3,4,8,7); (1,5,6,2)(3,4,8,7) gap> K8 := Subgroup(G8, [f8,l8]); Group([ (1,2,3,4)(5,6,7,8), (1,5,6,2)(3,4,8,7) ]) gap> Order(K8); 24 gap> Elements(K8); [ (), (2,4,5)(3,8,6), (2,5,4)(3,6,8), (1,2)(3,5)(4,6)(7,8), (1,2,3,4)(5,6,7,8), (1,2,6,5)(3,7,8,4), (1,3,6)(4,7,5), (1,3)(2,4)(5,7)(6,8), (1,3,8)(2,7,5), (1,4,3,2)(5,8,7,6), (1,4,8,5)(2,3,7,6), (1,4)(2,8)(3,5)(6,7), (1,5,6,2)(3,4,8,7), (1,5,8,4)(2,6,7,3), (1,5)(2,8)(3,7)(4,6), (1,6,3)(4,5,7), (1,6)(2,5)(3,8)(4,7), (1,6,8)(2,7,4), (1,7)(2,3)(4,6)(5,8), (1,7)(2,6)(3,5)(4,8), (1,7)(2,8)(3,4)(5,6), (1,8,6)(2,4,7), (1,8,3)(2,5,7), (1,8)(2,7)(3,6)(4,5) ] gap> f8*l8; (2,4,5)(3,8,6)
Exercise 1A.2 gap> f8^2; (1,3)(2,4)(5,7)(6,8) gap> f8^2*l8^2; (1,8)(2,7)(3,6)(4,5) gap> f8*l8*f8*l8; (2,5,4)(3,6,8)
gap> G6 := SymmetricGroup(6); Sym( [ 1 .. 6 ] ) gap> f6 := (1,5,6,3); (1,5,6,3) gap> l6 := (1,4,6,2); (1,4,6,2) gap> K6 := Subgroup(G6,[f6,l6]); Group([ (1,5,6,3), (1,4,6,2) ]) gap> Order(K6); 24
Exercise 1A.3 gap> f6^3; (1,3,6,5) gap> f6^3*l6^3; (1,3,4)(2,6,5)
Exercise 1A.5 gap> Factorization(K6,(1,3,4)(2,6,5)); x1^-1*x2^-1 gap> Factorization(K6,(1,6)(2,5)(3,4)); x1*x2*x1
Exercise 1A.7 gap> G4 := SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> f4 := (1,3,2,4); (1,3,2,4) gap> l4 := (1,2,4,3); (1,2,4,3) gap> K4 := Subgroup(G4,[f4,l4]); Group([ (1,3,2,4), (1,2,4,3) ]) gap> Order(K4); 24 gap> Elements(K4); [ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3) ]
gap> IsomorphismGroups(K8,K6); [ (1,2,3,4)(5,6,7,8), (1,5,6,2)(3,4,8,7) ] -> [ (1,3,6,5), (2,3,4,5) ] gap> IsomorphismGroups(K6,K4); [ (1,5,6,3), (1,4,6,2) ] -> [ (1,4,2,3), (1,4,3,2) ] gap> IsomorphismGroups(G6,G4); fail
Section 2: Direct Products of Groups
gap> C4 := CyclicGroup(IsPermGroup,4);
Group([ (1,2,3,4) ])
gap> S3 := SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> D := DirectProduct(S3,C4);
Group([ (1,2,3), (1,2), (4,5,6,7) ])
gap> Order(D);
24
gap> Read("orderFrequency");
gap> orderFrequency(D);
[ [ 1, 1 ], [ 2, 7 ], [ 3, 2 ], [ 4, 8 ], [ 6, 2 ], [ 12, 4 ] ]
Exercise 2.3 gap> D5 := DihedralGroup(IsPermGroup,10); Group([ (1,2,3,4,5), (2,5)(3,4) ]) gap> E1 := DirectProduct(D5,C4); Group([ (1,2,3,4,5), (2,5)(3,4), (6,7,8,9) ]) gap> orderFrequency(E1); [ [ 1, 1 ], [ 2, 11 ], [ 4, 12 ], [ 5, 4 ], [ 10, 4 ], [ 20, 8 ] ] gap> Order(E1); 40
Exercise 2.4 gap> D10 := DihedralGroup(IsPermGroup,20); Group([ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7) ]) gap> C2 := CyclicGroup(IsPermGroup,2); Group([ (1,2) ]) gap> E2 := DirectProduct(D10,C2); Group([ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7), (11,12) ]) gap> Order(E2); 40 gap> orderFrequency(E2); [ [ 1, 1 ], [ 2, 23 ], [ 5, 4 ], [ 10, 12 ] ]
Exercise 2.6 gap> D20 := DihedralGroup(IsPermGroup,40); Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10, 12) ]) gap> orderFrequency(D20); [ [ 1, 1 ], [ 2, 21 ], [ 4, 2 ], [ 5, 4 ], [ 10, 4 ], [ 20, 8 ] ]
gap> AllSmallGroups(40); [ <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators>, <pc group of size 40 with 4 generators> ] gap> List(AllSmallGroups(40), x -> orderFrequency(x)); [ [ [ 1, 1 ], [ 2, 1 ], [ 4, 2 ], [ 5, 4 ], [ 8, 20 ], [ 10, 4 ], [ 20, 8 ] ], [ [ 1, 1 ], [ 2, 1 ], [ 4, 2 ], [ 5, 4 ], [ 8, 4 ], [ 10, 4 ], [ 20, 8 ], [ 40, 16 ] ], [ [ 1, 1 ], [ 2, 1 ], [ 4, 10 ], [ 5, 4 ], [ 8, 20 ], [ 10, 4 ] ], [ [ 1, 1 ], [ 2, 1 ], [ 4, 22 ], [ 5, 4 ], [ 10, 4 ], [ 20, 8 ] ], [ [ 1, 1 ], [ 2, 11 ], [ 4, 12 ], [ 5, 4 ], [ 10, 4 ], [ 20, 8 ] ], [ [ 1, 1 ], [ 2, 21 ], [ 4, 2 ], [ 5, 4 ], [ 10, 4 ], [ 20, 8 ] ], [ [ 1, 1 ], [ 2, 3 ], [ 4, 20 ], [ 5, 4 ], [ 10, 12 ] ], [ [ 1, 1 ], [ 2, 13 ], [ 4, 10 ], [ 5, 4 ], [ 10, 12 ] ], [ [ 1, 1 ], [ 2, 3 ], [ 4, 4 ], [ 5, 4 ], [ 10, 12 ], [ 20, 16 ] ], [ [ 1, 1 ], [ 2, 5 ], [ 4, 2 ], [ 5, 4 ], [ 10, 20 ], [ 20, 8 ] ], [ [ 1, 1 ], [ 2, 1 ], [ 4, 6 ], [ 5, 4 ], [ 10, 4 ], [ 20, 24 ] ], [ [ 1, 1 ], [ 2, 11 ], [ 4, 20 ], [ 5, 4 ], [ 10, 4 ] ], [ [ 1, 1 ], [ 2, 23 ], [ 5, 4 ], [ 10, 12 ] ], [ [ 1, 1 ], [ 2, 7 ], [ 5, 4 ], [ 10, 28 ] ] ] gap> Size(AllSmallGroups(32)); 51 gap> Size(AllSmallGroups(64)); 267
Section 2 Appendix: Creating Functions for Later Use
gap> InputLogTo
> ("of");
InputLogTo: cannot log to of at
INPUT_LOG_TO( name );
called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> InputLogTo("of");
InputLogTo: cannot log to of at
INPUT_LOG_TO( name );
called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> quit;
gap> Read("freq");
gap> quit;
Section 3 : A Sylow Theorem
gap> Read("sylows");
gap> G:= SymmetricGroup(6);
Sym( [ 1 .. 6 ] )
gap> sylows(G,3);
[ Group([ (1,2,3), (4,5,6) ]), Group([ (1,2,4), (3,5,6) ]), Group([ (1,2,5), (3,4,6) ]), Group([ (1,2,6), (3,4,5) ]),
Group([ (1,3,4), (2,5,6) ]), Group([ (1,3,5), (2,4,6) ]), Group([ (1,3,6), (2,4,5) ]), Group([ (1,4,5), (2,3,6) ]),
Group([ (1,4,6), (2,3,5) ]), Group([ (1,5,6), (2,3,4) ]) ]
gap> sylows(G,2);
[ Group([ (1,2), (3,4), (1,3)(2,4), (5,6) ]), Group([ (1,2), (3,5), (1,3)(2,5), (4,6) ]),
Group([ (1,2), (3,6), (1,3)(2,6), (4,5) ]), Group([ (1,2), (4,5), (1,4)(2,5), (3,6) ]),
Group([ (1,2), (4,6), (1,4)(2,6), (3,5) ]), Group([ (1,2), (5,6), (1,5)(2,6), (3,4) ]),
Group([ (1,3), (2,4), (1,2)(3,4), (5,6) ]), Group([ (1,3), (2,5), (1,2)(3,5), (4,6) ]),
Group([ (1,3), (2,6), (1,2)(3,6), (4,5) ]), Group([ (1,3), (4,5), (1,4)(3,5), (2,6) ]),
Group([ (1,3), (4,6), (1,4)(3,6), (2,5) ]), Group([ (1,3), (5,6), (1,5)(3,6), (2,4) ]),
Group([ (1,4), (2,3), (1,2)(3,4), (5,6) ]), Group([ (1,4), (2,5), (1,2)(4,5), (3,6) ]),
Group([ (1,4), (2,6), (1,2)(4,6), (3,5) ]), Group([ (1,4), (3,5), (1,3)(4,5), (2,6) ]),
Group([ (1,4), (3,6), (1,3)(4,6), (2,5) ]), Group([ (1,4), (5,6), (1,5)(4,6), (2,3) ]),
Group([ (1,5), (2,3), (1,2)(3,5), (4,6) ]), Group([ (1,5), (2,4), (1,2)(4,5), (3,6) ]),
Group([ (1,5), (2,6), (1,2)(5,6), (3,4) ]), Group([ (1,5), (3,4), (1,3)(4,5), (2,6) ]),
Group([ (1,5), (3,6), (1,3)(5,6), (2,4) ]), Group([ (1,5), (4,6), (1,4)(5,6), (2,3) ]),
Group([ (1,6), (2,3), (1,2)(3,6), (4,5) ]), Group([ (1,6), (2,4), (1,2)(4,6), (3,5) ]),
Group([ (1,6), (2,5), (1,2)(5,6), (3,4) ]), Group([ (1,6), (3,4), (1,3)(4,6), (2,5) ]),
Group([ (1,6), (3,5), (1,3)(5,6), (2,4) ]), Group([ (1,6), (4,5), (1,4)(5,6), (2,3) ]),
Group([ (2,3), (4,5), (2,4)(3,5), (1,6) ]), Group([ (2,3), (4,6), (2,4)(3,6), (1,5) ]),
Group([ (2,3), (5,6), (2,5)(3,6), (1,4) ]), Group([ (2,4), (3,5), (2,3)(4,5), (1,6) ]),
Group([ (2,4), (3,6), (2,3)(4,6), (1,5) ]), Group([ (2,4), (5,6), (2,5)(4,6), (1,3) ]),
Group([ (2,5), (3,4), (2,3)(4,5), (1,6) ]), Group([ (2,5), (3,6), (2,3)(5,6), (1,4) ]),
Group([ (2,5), (4,6), (2,4)(5,6), (1,3) ]), Group([ (2,6), (3,4), (2,3)(4,6), (1,5) ]),
Group([ (2,6), (3,5), (2,3)(5,6), (1,4) ]), Group([ (2,6), (4,5), (2,4)(5,6), (1,3) ]),
Group([ (3,4), (5,6), (3,5)(4,6), (1,2) ]), Group([ (3,5), (4,6), (3,4)(5,6), (1,2) ]),
Group([ (3,6), (4,5), (3,4)(5,6), (1,2) ]) ]
Exercise 3.2 gap> G:= SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> sylows(G,2); [ Group([ (1,2), (3,4), (1,3)(2,4) ]), Group([ (1,3), (2,4), (1,2)(3,4) ]), Group([ (1,4), (2,3), (1,2)(3,4) ]) ] gap> sylows(G,3); [ Group([ (1,2,3) ]), Group([ (2,3,4) ]), Group([ (1,3,4) ]), Group([ (1,4,2) ]) ]
Exercise 3.3 gap> G:= AlternatingGroup(6); Alt( [ 1 .. 6 ] ) gap> Size(sylows(G,2)); 45 gap> Size(sylows(G,3)); 10 gap> Size(sylows(G,5)); 36
Exercise 3.4 gap> G:= SymmetricGroup(7); Sym( [ 1 .. 7 ] ) gap> Size(sylows(G,2)); 315 gap> Size(sylows(G,3)); 70 gap> Size(sylows(G,5)); 126 gap> Size(sylows(G,7)); 120
Exercise 3.5 gap> C:= CyclicGroup(60); <pc group of size 60 with 4 generators> gap> Size(sylows(C,2)); 1 gap> Size(sylows(C,3)); 1 gap> Size(sylows(C,5)); 1 gap> sylows(G,5); [ Group([ (1,2,3,4,5) ]), Group([ (1,2,3,4,6) ]), Group([ (1,2,3,4,7) ]), Group([ (1,2,3,5,4) ]), Group([ (1,2,3,5,6) ]), Group([ (1,2,3,5,7) ]), Group([ (1,2,3,6,4) ]), Group([ (1,2,3,6,5) ]), Group([ (1,2,3,6,7) ]), Group([ (1,2,3,7,4) ]), Group([ (1,2,3,7,5) ]), Group([ (1,2,3,7,6) ]), Group([ (1,2,4,3,5) ]), Group([ (1,2,4,3,6) ]), Group([ (1,2,4,3,7) ]), Group([ (1,2,4,5,3) ]), Group([ (1,2,4,5,6) ]), Group([ (1,2,4,5,7) ]), Group([ (1,2,4,6,3) ]), Group([ (1,2,4,6,5) ]), Group([ (1,2,4,6,7) ]), Group([ (1,2,4,7,3) ]), Group([ (1,2,4,7,5) ]), Group([ (1,2,4,7,6) ]), Group([ (1,2,5,3,4) ]), Group([ (1,2,5,3,6) ]), Group([ (1,2,5,3,7) ]), Group([ (1,2,5,4,3) ]), Group([ (1,2,5,4,6) ]), Group([ (1,2,5,4,7) ]), Group([ (1,2,5,6,3) ]), Group([ (1,2,5,6,4) ]), Group([ (1,2,5,6,7) ]), Group([ (1,2,5,7,3) ]), Group([ (1,2,5,7,4) ]), Group([ (1,2,5,7,6) ]), Group([ (1,2,6,3,4) ]), Group([ (1,2,6,3,5) ]), Group([ (1,2,6,3,7) ]), Group([ (1,2,6,4,3) ]), Group([ (1,2,6,4,5) ]), Group([ (1,2,6,4,7) ]), Group([ (1,2,6,5,3) ]), Group([ (1,2,6,5,4) ]), Group([ (1,2,6,5,7) ]), Group([ (1,2,6,7,3) ]), Group([ (1,2,6,7,4) ]), Group([ (1,2,6,7,5) ]), Group([ (1,2,7,3,4) ]), Group([ (1,2,7,3,5) ]), Group([ (1,2,7,3,6) ]), Group([ (1,2,7,4,3) ]), Group([ (1,2,7,4,5) ]), Group([ (1,2,7,4,6) ]), Group([ (1,2,7,5,3) ]), Group([ (1,2,7,5,4) ]), Group([ (1,2,7,5,6) ]), Group([ (1,2,7,6,3) ]), Group([ (1,2,7,6,4) ]), Group([ (1,2,7,6,5) ]), Group([ (1,3,4,5,6) ]), Group([ (1,3,4,5,7) ]), Group([ (1,3,4,6,5) ]), Group([ (1,3,4,6,7) ]), Group([ (1,3,4,7,5) ]), Group([ (1,3,4,7,6) ]), Group([ (1,3,5,4,6) ]), Group([ (1,3,5,4,7) ]), Group([ (1,3,5,6,4) ]), Group([ (1,3,5,6,7) ]), Group([ (1,3,5,7,4) ]), Group([ (1,3,5,7,6) ]), Group([ (1,3,6,4,5) ]), Group([ (1,3,6,4,7) ]), Group([ (1,3,6,5,4) ]), Group([ (1,3,6,5,7) ]), Group([ (1,3,6,7,4) ]), Group([ (1,3,6,7,5) ]), Group([ (1,3,7,4,5) ]), Group([ (1,3,7,4,6) ]), Group([ (1,3,7,5,4) ]), Group([ (1,3,7,5,6) ]), Group([ (1,3,7,6,4) ]), Group([ (1,3,7,6,5) ]), Group([ (1,4,5,6,7) ]), Group([ (1,4,5,7,6) ]), Group([ (1,4,6,5,7) ]), Group([ (1,4,6,7,5) ]), Group([ (1,4,7,5,6) ]), Group([ (1,4,7,6,5) ]), Group([ (2,3,4,5,6) ]), Group([ (2,3,4,5,7) ]), Group([ (2,3,4,6,5) ]), Group([ (2,3,4,6,7) ]), Group([ (2,3,4,7,5) ]), Group([ (2,3,4,7,6) ]), Group([ (2,3,5,4,6) ]), Group([ (2,3,5,4,7) ]), Group([ (2,3,5,6,4) ]), Group([ (2,3,5,6,7) ]), Group([ (2,3,5,7,4) ]), Group([ (2,3,5,7,6) ]), Group([ (2,3,6,4,5) ]), Group([ (2,3,6,4,7) ]), Group([ (2,3,6,5,4) ]), Group([ (2,3,6,5,7) ]), Group([ (2,3,6,7,4) ]), Group([ (2,3,6,7,5) ]), Group([ (2,3,7,4,5) ]), Group([ (2,3,7,4,6) ]), Group([ (2,3,7,5,4) ]), Group([ (2,3,7,5,6) ]), Group([ (2,3,7,6,4) ]), Group([ (2,3,7,6,5) ]), Group([ (2,4,5,6,7) ]), Group([ (2,4,5,7,6) ]), Group([ (2,4,6,5,7) ]), Group([ (2,4,6,7,5) ]), Group([ (2,4,7,5,6) ]), Group([ (2,4,7,6,5) ]), Group([ (3,4,5,6,7) ]), Group([ (3,4,5,7,6) ]), Group([ (3,4,6,5,7) ]), Group([ (3,4,6,7,5) ]), Group([ (3,4,7,5,6) ]), Group([ (3,4,7,6,5) ]) ] gap> a:= sylows(G,5); [ Group([ (1,2,3,4,5) ]), Group([ (1,2,3,4,6) ]), Group([ (1,2,3,4,7) ]), Group([ (1,2,3,5,4) ]), Group([ (1,2,3,5,6) ]), Group([ (1,2,3,5,7) ]), Group([ (1,2,3,6,4) ]), Group([ (1,2,3,6,5) ]), Group([ (1,2,3,6,7) ]), Group([ (1,2,3,7,4) ]), Group([ (1,2,3,7,5) ]), Group([ (1,2,3,7,6) ]), Group([ (1,2,4,3,5) ]), Group([ (1,2,4,3,6) ]), Group([ (1,2,4,3,7) ]), Group([ (1,2,4,5,3) ]), Group([ (1,2,4,5,6) ]), Group([ (1,2,4,5,7) ]), Group([ (1,2,4,6,3) ]), Group([ (1,2,4,6,5) ]), Group([ (1,2,4,6,7) ]), Group([ (1,2,4,7,3) ]), Group([ (1,2,4,7,5) ]), Group([ (1,2,4,7,6) ]), Group([ (1,2,5,3,4) ]), Group([ (1,2,5,3,6) ]), Group([ (1,2,5,3,7) ]), Group([ (1,2,5,4,3) ]), Group([ (1,2,5,4,6) ]), Group([ (1,2,5,4,7) ]), Group([ (1,2,5,6,3) ]), Group([ (1,2,5,6,4) ]), Group([ (1,2,5,6,7) ]), Group([ (1,2,5,7,3) ]), Group([ (1,2,5,7,4) ]), Group([ (1,2,5,7,6) ]), Group([ (1,2,6,3,4) ]), Group([ (1,2,6,3,5) ]), Group([ (1,2,6,3,7) ]), Group([ (1,2,6,4,3) ]), Group([ (1,2,6,4,5) ]), Group([ (1,2,6,4,7) ]), Group([ (1,2,6,5,3) ]), Group([ (1,2,6,5,4) ]), Group([ (1,2,6,5,7) ]), Group([ (1,2,6,7,3) ]), Group([ (1,2,6,7,4) ]), Group([ (1,2,6,7,5) ]), Group([ (1,2,7,3,4) ]), Group([ (1,2,7,3,5) ]), Group([ (1,2,7,3,6) ]), Group([ (1,2,7,4,3) ]), Group([ (1,2,7,4,5) ]), Group([ (1,2,7,4,6) ]), Group([ (1,2,7,5,3) ]), Group([ (1,2,7,5,4) ]), Group([ (1,2,7,5,6) ]), Group([ (1,2,7,6,3) ]), Group([ (1,2,7,6,4) ]), Group([ (1,2,7,6,5) ]), Group([ (1,3,4,5,6) ]), Group([ (1,3,4,5,7) ]), Group([ (1,3,4,6,5) ]), Group([ (1,3,4,6,7) ]), Group([ (1,3,4,7,5) ]), Group([ (1,3,4,7,6) ]), Group([ (1,3,5,4,6) ]), Group([ (1,3,5,4,7) ]), Group([ (1,3,5,6,4) ]), Group([ (1,3,5,6,7) ]), Group([ (1,3,5,7,4) ]), Group([ (1,3,5,7,6) ]), Group([ (1,3,6,4,5) ]), Group([ (1,3,6,4,7) ]), Group([ (1,3,6,5,4) ]), Group([ (1,3,6,5,7) ]), Group([ (1,3,6,7,4) ]), Group([ (1,3,6,7,5) ]), Group([ (1,3,7,4,5) ]), Group([ (1,3,7,4,6) ]), Group([ (1,3,7,5,4) ]), Group([ (1,3,7,5,6) ]), Group([ (1,3,7,6,4) ]), Group([ (1,3,7,6,5) ]), Group([ (1,4,5,6,7) ]), Group([ (1,4,5,7,6) ]), Group([ (1,4,6,5,7) ]), Group([ (1,4,6,7,5) ]), Group([ (1,4,7,5,6) ]), Group([ (1,4,7,6,5) ]), Group([ (2,3,4,5,6) ]), Group([ (2,3,4,5,7) ]), Group([ (2,3,4,6,5) ]), Group([ (2,3,4,6,7) ]), Group([ (2,3,4,7,5) ]), Group([ (2,3,4,7,6) ]), Group([ (2,3,5,4,6) ]), Group([ (2,3,5,4,7) ]), Group([ (2,3,5,6,4) ]), Group([ (2,3,5,6,7) ]), Group([ (2,3,5,7,4) ]), Group([ (2,3,5,7,6) ]), Group([ (2,3,6,4,5) ]), Group([ (2,3,6,4,7) ]), Group([ (2,3,6,5,4) ]), Group([ (2,3,6,5,7) ]), Group([ (2,3,6,7,4) ]), Group([ (2,3,6,7,5) ]), Group([ (2,3,7,4,5) ]), Group([ (2,3,7,4,6) ]), Group([ (2,3,7,5,4) ]), Group([ (2,3,7,5,6) ]), Group([ (2,3,7,6,4) ]), Group([ (2,3,7,6,5) ]), Group([ (2,4,5,6,7) ]), Group([ (2,4,5,7,6) ]), Group([ (2,4,6,5,7) ]), Group([ (2,4,6,7,5) ]), Group([ (2,4,7,5,6) ]), Group([ (2,4,7,6,5) ]), Group([ (3,4,5,6,7) ]), Group([ (3,4,5,7,6) ]), Group([ (3,4,6,5,7) ]), Group([ (3,4,6,7,5) ]), Group([ (3,4,7,5,6) ]), Group([ (3,4,7,6,5) ]) ] gap> a[1]; Group([ (1,2,3,4,5) ]) gap> Elements(a[1]); [ (), (1,2,3,4,5), (1,3,5,2,4), (1,4,2,5,3), (1,5,4,3,2) ] gap> a[4]; Group([ (1,2,3,5,4) ]) gap> Elements(a[4]); [ (), (1,2,3,5,4), (1,3,4,2,5), (1,4,5,3,2), (1,5,2,4,3) ] gap> b:= Elements(G); [ (), (6,7), (5,6), (5,6,7), (5,7,6), (5,7), (4,5), (4,5)(6,7), (4,5,6), (4,5,6,7), (4,5,7,6), (4,5,7), (4,6,5), (4,6,7,5), (4,6), (4,6,7), (4,6)(5,7), (4,6,5,7), (4,7,6,5), (4,7,5), (4,7,6), (4,7), (4,7,5,6), (4,7)(5,6), (3,4), (3,4)(6,7), (3,4)(5,6), (3,4)(5,6,7), (3,4)(5,7,6), (3,4)(5,7), (3,4,5), (3,4,5)(6,7), (3,4,5,6), (3,4,5,6,7), (3,4,5,7,6), (3,4,5,7), (3,4,6,5), (3,4,6,7,5), (3,4,6), (3,4,6,7), (3,4,6)(5,7), (3,4,6,5,7), (3,4,7,6,5), (3,4,7,5), (3,4,7,6), (3,4,7), (3,4,7,5,6), (3,4,7)(5,6), (3,5,4), (3,5,4)(6,7), (3,5,6,4), (3,5,6,7,4), (3,5,7,6,4), (3,5,7,4), (3,5), (3,5)(6,7), (3,5,6), (3,5,6,7), (3,5,7,6), (3,5,7), (3,5)(4,6), (3,5)(4,6,7), (3,5,4,6), (3,5,4,6,7), (3,5,7,4,6), (3,5,7)(4,6), (3,5)(4,7,6), (3,5)(4,7), (3,5,4,7,6), (3,5,4,7), (3,5,6)(4,7), (3,5,6,4,7), (3,6,5,4), (3,6,7,5,4), (3,6,4), (3,6,7,4), (3,6,4)(5,7), (3,6,5,7,4), (3,6,5), (3,6,7,5), (3,6), (3,6,7), (3,6)(5,7), (3,6,5,7), (3,6,4,5), (3,6,7,4,5), (3,6)(4,5), (3,6,7)(4,5), (3,6)(4,5,7), (3,6,4,5,7), (3,6,4,7,5), (3,6,5)(4,7), (3,6)(4,7,5), (3,6,5,4,7), (3,6)(4,7), (3,6,4,7), (3,7,6,5,4), (3,7,5,4), (3,7,6,4), (3,7,4), (3,7,5,6,4), (3,7,4)(5,6), (3,7,6,5), (3,7,5), (3,7,6), (3,7), (3,7,5,6), (3,7)(5,6), (3,7,6,4,5), (3,7,4,5), (3,7,6)(4,5), (3,7)(4,5), (3,7,4,5,6), (3,7)(4,5,6), (3,7,5)(4,6), (3,7,4,6,5), (3,7,5,4,6), (3,7)(4,6,5), (3,7,4,6), (3,7)(4,6), (2,3), (2,3)(6,7), (2,3)(5,6), (2,3)(5,6,7), (2,3)(5,7,6), (2,3)(5,7), (2,3)(4,5), (2,3)(4,5)(6,7), (2,3)(4,5,6), (2,3)(4,5,6,7), (2,3)(4,5,7,6), (2,3)(4,5,7), (2,3)(4,6,5), (2,3)(4,6,7,5), (2,3)(4,6), (2,3)(4,6,7), (2,3)(4,6)(5,7), (2,3)(4,6,5,7), (2,3)(4,7,6,5), (2,3)(4,7,5), (2,3)(4,7,6), (2,3)(4,7), (2,3)(4,7,5,6), (2,3)(4,7)(5,6), (2,3,4), (2,3,4)(6,7), (2,3,4)(5,6), (2,3,4)(5,6,7), (2,3,4)(5,7,6), (2,3,4)(5,7), (2,3,4,5), (2,3,4,5)(6,7), (2,3,4,5,6), (2,3,4,5,6,7), (2,3,4,5,7,6), (2,3,4,5,7), (2,3,4,6,5), (2,3,4,6,7,5), (2,3,4,6), (2,3,4,6,7), (2,3,4,6)(5,7), (2,3,4,6,5,7), (2,3,4,7,6,5), (2,3,4,7,5), (2,3,4,7,6), (2,3,4,7), (2,3,4,7,5,6), (2,3,4,7)(5,6), (2,3,5,4), (2,3,5,4)(6,7), (2,3,5,6,4), (2,3,5,6,7,4), (2,3,5,7,6,4), (2,3,5,7,4), (2,3,5), (2,3,5)(6,7), (2,3,5,6), (2,3,5,6,7), (2,3,5,7,6), (2,3,5,7), (2,3,5)(4,6), (2,3,5)(4,6,7), (2,3,5,4,6), (2,3,5,4,6,7), (2,3,5,7,4,6), (2,3,5,7)(4,6), (2,3,5)(4,7,6), (2,3,5)(4,7), (2,3,5,4,7,6), (2,3,5,4,7), (2,3,5,6)(4,7), (2,3,5,6,4,7), (2,3,6,5,4), (2,3,6,7,5,4), (2,3,6,4), (2,3,6,7,4), (2,3,6,4)(5,7), (2,3,6,5,7,4), (2,3,6,5), (2,3,6,7,5), (2,3,6), (2,3,6,7), (2,3,6)(5,7), (2,3,6,5,7), (2,3,6,4,5), (2,3,6,7,4,5), (2,3,6)(4,5), (2,3,6,7)(4,5), (2,3,6)(4,5,7), (2,3,6,4,5,7), (2,3,6,4,7,5), (2,3,6,5)(4,7), (2,3,6)(4,7,5), (2,3,6,5,4,7), (2,3,6)(4,7), (2,3,6,4,7), (2,3,7,6,5,4), (2,3,7,5,4), (2,3,7,6,4), (2,3,7,4), (2,3,7,5,6,4), (2,3,7,4)(5,6), (2,3,7,6,5), (2,3,7,5), (2,3,7,6), (2,3,7), (2,3,7,5,6), (2,3,7)(5,6), (2,3,7,6,4,5), (2,3,7,4,5), (2,3,7,6)(4,5), (2,3,7)(4,5), (2,3,7,4,5,6), (2,3,7)(4,5,6), (2,3,7,5)(4,6), (2,3,7,4,6,5), (2,3,7,5,4,6), (2,3,7)(4,6,5), (2,3,7,4,6), (2,3,7)(4,6), (2,4,3), (2,4,3)(6,7), (2,4,3)(5,6), (2,4,3)(5,6,7), (2,4,3)(5,7,6), (2,4,3)(5,7), (2,4,5,3), (2,4,5,3)(6,7), (2,4,5,6,3), (2,4,5,6,7,3), (2,4,5,7,6,3), (2,4,5,7,3), (2,4,6,5,3), (2,4,6,7,5,3), (2,4,6,3), (2,4,6,7,3), (2,4,6,3)(5,7), (2,4,6,5,7,3), (2,4,7,6,5,3), (2,4,7,5,3), (2,4,7,6,3), (2,4,7,3), (2,4,7,5,6,3), (2,4,7,3)(5,6), (2,4), (2,4)(6,7), (2,4)(5,6), (2,4)(5,6,7), (2,4)(5,7,6), (2,4)(5,7), (2,4,5), (2,4,5)(6,7), (2,4,5,6), (2,4,5,6,7), (2,4,5,7,6), (2,4,5,7), (2,4,6,5), (2,4,6,7,5), (2,4,6), (2,4,6,7), (2,4,6)(5,7), (2,4,6,5,7), (2,4,7,6,5), (2,4,7,5), (2,4,7,6), (2,4,7), (2,4,7,5,6), (2,4,7)(5,6), (2,4)(3,5), (2,4)(3,5)(6,7), (2,4)(3,5,6), (2,4)(3,5,6,7), (2,4)(3,5,7,6), (2,4)(3,5,7), (2,4,3,5), (2,4,3,5)(6,7), (2,4,3,5,6), (2,4,3,5,6,7), (2,4,3,5,7,6), (2,4,3,5,7), (2,4,6,3,5), (2,4,6,7,3,5), (2,4,6)(3,5), (2,4,6,7)(3,5), (2,4,6)(3,5,7), (2,4,6,3,5,7), (2,4,7,6,3,5), (2,4,7,3,5), (2,4,7,6)(3,5), (2,4,7)(3,5), (2,4,7,3,5,6), (2,4,7)(3,5,6), (2,4)(3,6,5), (2,4)(3,6,7,5), (2,4)(3,6), (2,4)(3,6,7), (2,4)(3,6)(5,7), (2,4)(3,6,5,7), (2,4,3,6,5), (2,4,3,6,7,5), (2,4,3,6), (2,4,3,6,7), (2,4,3,6)(5,7), (2,4,3,6,5,7), (2,4,5)(3,6), (2,4,5)(3,6,7), (2,4,5,3,6), (2,4,5,3,6,7), (2,4,5,7,3,6), (2,4,5,7)(3,6), (2,4,7,5)(3,6), (2,4,7,3,6,5), (2,4,7,5,3,6), (2,4,7)(3,6,5), (2,4,7,3,6), (2,4,7)(3,6), (2,4)(3,7,6,5), (2,4)(3,7,5), (2,4)(3,7,6), (2,4)(3,7), (2,4)(3,7,5,6), (2,4)(3,7)(5,6), (2,4,3,7,6,5), (2,4,3,7,5), 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(1,7,4)(2,5,6,3), (1,7,3,2,5,6,4), (1,7,6,4,3,2,5), (1,7,4,3,2,5), (1,7,6)(2,5,4,3), (1,7)(2,5,4,3), (1,7,4,3,2,5,6), (1,7)(2,5,6,4,3), (1,7,6,3,2,5), (1,7,3,2,5), (1,7,6)(2,5,3), (1,7)(2,5,3), (1,7,3,2,5,6), (1,7)(2,5,6,3), (1,7,4,6,3,2,5), (1,7,3,2,5)(4,6), (1,7,4,6)(2,5,3), (1,7)(2,5,3)(4,6), (1,7,3,2,5,4,6), (1,7)(2,5,4,6,3), (1,7,6,4)(2,5), (1,7,4)(2,5), (1,7,6,2,5,4), (1,7,2,5,4), (1,7,4)(2,5,6), (1,7,2,5,6,4), (1,7,6,4,2,5), (1,7,4,2,5), (1,7,6)(2,5,4), (1,7)(2,5,4), (1,7,4,2,5,6), (1,7)(2,5,6,4), (1,7,6,2,5), (1,7,2,5), (1,7,6)(2,5), (1,7)(2,5), (1,7,2,5,6), (1,7)(2,5,6), (1,7,4,6,2,5), (1,7,2,5)(4,6), (1,7,4,6)(2,5), (1,7)(2,5)(4,6), (1,7,2,5,4,6), (1,7)(2,5,4,6), (1,7,6,3,4)(2,5), (1,7,3,4)(2,5), (1,7,6,2,5,3,4), (1,7,2,5,3,4), (1,7,3,4)(2,5,6), (1,7,2,5,6,3,4), (1,7,6,3,4,2,5), (1,7,3,4,2,5), (1,7,6)(2,5,3,4), (1,7)(2,5,3,4), (1,7,3,4,2,5,6), (1,7)(2,5,6,3,4), (1,7,6,2,5)(3,4), (1,7,2,5)(3,4), (1,7,6)(2,5)(3,4), (1,7)(2,5)(3,4), (1,7,2,5,6)(3,4), (1,7)(2,5,6)(3,4), (1,7,3,4,6,2,5), (1,7,2,5)(3,4,6), (1,7,3,4,6)(2,5), (1,7)(2,5)(3,4,6), (1,7,2,5,3,4,6), (1,7)(2,5,3,4,6), (1,7,4)(2,5)(3,6), (1,7,3,6,4)(2,5), (1,7,4)(2,5,3,6), (1,7,2,5,3,6,4), (1,7,3,6,2,5,4), (1,7,2,5,4)(3,6), (1,7,4,2,5)(3,6), (1,7,3,6,4,2,5), (1,7,4,2,5,3,6), (1,7)(2,5,3,6,4), (1,7,3,6)(2,5,4), (1,7)(2,5,4)(3,6), (1,7,4,3,6,2,5), (1,7,2,5)(3,6,4), (1,7,4,3,6)(2,5), (1,7)(2,5)(3,6,4), (1,7,2,5,4,3,6), (1,7)(2,5,4,3,6), (1,7,3,6,2,5), (1,7,2,5)(3,6), (1,7,3,6)(2,5), (1,7)(2,5)(3,6), (1,7,2,5,3,6), (1,7)(2,5,3,6), (1,7,5,3)(2,6,4), (1,7,4,2,6,5,3), (1,7,5,4,2,6,3), (1,7,3)(2,6,5,4), (1,7,4,2,6,3), (1,7,3)(2,6,4), (1,7,5,2,6,4,3), (1,7,4,3)(2,6,5), (1,7,5,4,3)(2,6), (1,7,2,6,5,4,3), (1,7,4,3)(2,6), (1,7,2,6,4,3), (1,7,5,2,6,3), (1,7,3)(2,6,5), (1,7,5,3)(2,6), (1,7,2,6,5,3), (1,7,3)(2,6), (1,7,2,6,3), (1,7,4,5,2,6,3), (1,7,3)(2,6,4,5), (1,7,4,5,3)(2,6), (1,7,2,6,4,5,3), (1,7,3)(2,6)(4,5), (1,7,2,6,3)(4,5), (1,7,5,3,2,6,4), (1,7,4)(2,6,5,3), (1,7,5,4)(2,6,3), (1,7,3,2,6,5,4), (1,7,4)(2,6,3), (1,7,3,2,6,4), (1,7,5)(2,6,4,3), (1,7,4,3,2,6,5), (1,7,5,4,3,2,6), (1,7)(2,6,5,4,3), (1,7,4,3,2,6), (1,7)(2,6,4,3), (1,7,5)(2,6,3), (1,7,3,2,6,5), (1,7,5,3,2,6), (1,7)(2,6,5,3), (1,7,3,2,6), (1,7)(2,6,3), (1,7,4,5)(2,6,3), (1,7,3,2,6,4,5), (1,7,4,5,3,2,6), (1,7)(2,6,4,5,3), (1,7,3,2,6)(4,5), (1,7)(2,6,3)(4,5), (1,7,5,2,6,4), (1,7,4)(2,6,5), (1,7,5,4)(2,6), (1,7,2,6,5,4), (1,7,4)(2,6), (1,7,2,6,4), (1,7,5)(2,6,4), (1,7,4,2,6,5), (1,7,5,4,2,6), (1,7)(2,6,5,4), (1,7,4,2,6), (1,7)(2,6,4), (1,7,5)(2,6), (1,7,2,6,5), (1,7,5,2,6), (1,7)(2,6,5), (1,7,2,6), (1,7)(2,6), (1,7,4,5)(2,6), (1,7,2,6,4,5), (1,7,4,5,2,6), (1,7)(2,6,4,5), (1,7,2,6)(4,5), (1,7)(2,6)(4,5), (1,7,5,2,6,3,4), (1,7,3,4)(2,6,5), (1,7,5,3,4)(2,6), (1,7,2,6,5,3,4), (1,7,3,4)(2,6), (1,7,2,6,3,4), (1,7,5)(2,6,3,4), (1,7,3,4,2,6,5), (1,7,5,3,4,2,6), (1,7)(2,6,5,3,4), (1,7,3,4,2,6), (1,7)(2,6,3,4), (1,7,5)(2,6)(3,4), (1,7,2,6,5)(3,4), (1,7,5,2,6)(3,4), (1,7)(2,6,5)(3,4), (1,7,2,6)(3,4), (1,7)(2,6)(3,4), (1,7,3,4,5)(2,6), (1,7,2,6,3,4,5), (1,7,3,4,5,2,6), (1,7)(2,6,3,4,5), (1,7,2,6)(3,4,5), (1,7)(2,6)(3,4,5), (1,7,4)(2,6,3,5), (1,7,3,5,2,6,4), (1,7,4)(2,6)(3,5), (1,7,2,6,4)(3,5), (1,7,3,5,4)(2,6), (1,7,2,6,3,5,4), (1,7,4,2,6,3,5), (1,7,3,5)(2,6,4), (1,7,4,2,6)(3,5), (1,7)(2,6,4)(3,5), (1,7,3,5,4,2,6), (1,7)(2,6,3,5,4), (1,7,4,3,5)(2,6), (1,7,2,6,4,3,5), (1,7,4,3,5,2,6), (1,7)(2,6,4,3,5), (1,7,2,6)(3,5,4), (1,7)(2,6)(3,5,4), (1,7,3,5)(2,6), (1,7,2,6,3,5), (1,7,3,5,2,6), (1,7)(2,6,3,5), (1,7,2,6)(3,5), (1,7)(2,6)(3,5) ] gap> b:= Elements(G);; gap> b[2]; (6,7) gap> LogTo();Back to Home Page
This PREP workshop is made possible by the NSF grant DUE: 0341481