Section 1, continued

gap> G:= DihedralGroup(8);

gap> H:= Subgroup(G, List(Elements(G), x -> x^2)));
Syntax error: ; expected
H:= Subgroup(G, List(Elements(G), x -> x^2)));
                                            ^
gap> H:= Subgroup(G, List(Elements(G), x -> x^2));
Group([  of ...,  of ..., f3,  of ..., 
   of ...,  of ..., f3,  of ... ])
gap> Order(H);
2
gap> Elements(H);
[  of ..., f3 ]
gap> G:= DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> H:= Subgroup(G, List(Elements(G), x -> x^2));
Group([ (), (), (), (1,3)(2,4), (), (), (1,3)(2,4), () ])
gap> Elements(H);
[ (), (1,3)(2,4) ]
gap> Order(Group([ (1,2), (1,2,3,4)]));
24
gap> Order(Group([ (1,2), (1,2,3,4)])); Factorial(4);
24
24
gap> Order(Group([ (1,2), (1,2,3,4,5)])); Factorial(5);
120
120
gap> Order(Group([ (1,3), (1,2,3,4,5)])); Factorial(5);
120
120
gap> Order(Group([ (1,4), (1,2,3,4,5)])); Factorial(5);
120
120
gap> g:= Group([(1,3), (1,4,5)]);
Group([ (1,3), (1,4,5) ])
gap> KnownAttributesOfObject(g);
[ "LargestMovedPoint", "GeneratorsOfMagmaWithInverses", 
  "MultiplicativeNeutralElement" ]
gap> HELP("LargestMovedPoint");
Help: Showing `Reference: LargestMovedPoint'
gap> Order(g);
24
gap> KnownAttributesOfObject(g);
[ "Size", "OneImmutable", "Order", "LargestMovedPoint", "NrMovedPoints", 
  "MovedPoints", "GeneratorsOfMagmaWithInverses", 
  "MultiplicativeNeutralElement", "Pcgs", "GeneralizedPcgs", "StabChainMutable",
  "StabChainOptions" ]
gap> LargestMovedPoint(g);
5
gap> NrMovedPoints(g);
4

Section 2

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> f8*l8;
(2,4,5)(3,8,6)
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> f8^2;
(1,3)(2,4)(5,7)(6,8)
gap> f8^2*l8^2;
(1,8)(2,7)(3,6)(4,5)
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
gap> f6^2;
(1,6)(3,5)
gap> f6*l6*f6*l6*f6;
(1,2,6,4)
gap> Factorization(G6,(1,3,4)(2,6,5));
x1*x2*x1^-2*x2*x1*x2*x1^3
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> l^2;
Variable: 'l' must have a value

gap> l4^2;
(1,4)(2,3)
gap> IsomorphismGroups(K8,K6);
[ (1,2,3,4)(5,6,7,8), (1,5,6,2)(3,4,8,7) ] -> [ (1,2,6,4), (2,5,4,3) ]
gap> IsomorphismGroups(K8,K4);
[ (1,2,3,4)(5,6,7,8), (1,5,6,2)(3,4,8,7) ] -> [ (1,3,4,2), (1,2,3,4) ]
gap> IsomorphismGroups(K6,K4);
[ (1,5,6,3), (1,4,6,2) ] -> [ (1,4,2,3), (1,2,4,3) ]

Section 3

gap> c6:=CyclicGroup(IsPermGroup,6);
Group([ (1,2,3,4,5,6) ])
gap> Elements(c6);
[ (), (1,2,3,4,5,6), (1,3,5)(2,4,6), (1,4)(2,5)(3,6), (1,5,3)(2,6,4), 
  (1,6,5,4,3,2) ]
gap> a:=c6.1;
(1,2,3,4,5,6)
gap> Elements(Subgroup(c6,[a^2]));
[ (), (1,3,5)(2,4,6), (1,5,3)(2,6,4) ]
gap> Elements(Subgroup(c6,[a^2,a^3]));
[ (), (1,2,3,4,5,6), (1,3,5)(2,4,6), (1,4)(2,5)(3,6), (1,5,3)(2,6,4), 
  (1,6,5,4,3,2) ]
gap> d4:=DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> Elements(d4);
[ (), (2,4), (1,2)(3,4), (1,2,3,4), (1,3), (1,3)(2,4), (1,4,3,2), (1,4)(2,3) ]
gap> Elements(Subgroup(d4,[(2,4)]));
[ (), (2,4) ]
gap> Elements(Subgroup(d4,[(1,2)(3,4)]));
[ (), (1,2)(3,4) ]
gap> Elements(Subgroup(d4,[(1,2,3,4)]));
[ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) ]
gap> Elements(Subgroup(d4,[(1,2)(3,4),(1,3)(2,4)]));
[ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ]
gap> c30:=CyclicGroup(IsPermGroup,30);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
    27,28,29,30) ])
gap> a:=c30.1;
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
30)
gap> Elements(c30,[a^4,a^6]);
Function: number of arguments must be 1 (not 2)
not in any function
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can replace the argument list  via 'return ;' to continue
brk> quit;
gap> Elements(Subgroup(c30,[a^4,a^6]));
[ (), (1,3,5,7,9,11,13,15,17,19,21,23,25,27,29)(2,4,6,8,10,12,14,16,18,20,22,24,
    26,28,30), (1,5,9,13,17,21,25,29,3,7,11,15,19,23,27)(2,6,10,14,18,22,26,30,
    4,8,12,16,20,24,28), (1,7,13,19,25)(2,8,14,20,26)(3,9,15,21,27)(4,10,16,22,
    28)(5,11,17,23,29)(6,12,18,24,30), (1,9,17,25,3,11,19,27,5,13,21,29,7,15,
    23)(2,10,18,26,4,12,20,28,6,14,22,30,8,16,24), 
  (1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,28)(9,19,
    29)(10,20,30), (1,13,25,7,19)(2,14,26,8,20)(3,15,27,9,21)(4,16,28,10,22)(5,
    17,29,11,23)(6,18,30,12,24), (1,15,29,13,27,11,25,9,23,7,21,5,19,3,17)(2,16,
    30,14,28,12,26,10,24,8,22,6,20,4,18), (1,17,3,19,5,21,7,23,9,25,11,27,13,29,
    15)(2,18,4,20,6,22,8,24,10,26,12,28,14,30,16), 
  (1,19,7,25,13)(2,20,8,26,14)(3,21,9,27,15)(4,22,10,28,16)(5,23,11,29,17)(6,24,
    12,30,18), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,
    17)(8,28,18)(9,29,19)(10,30,20), (1,23,15,7,29,21,13,5,27,19,11,3,25,17,
    9)(2,24,16,8,30,22,14,6,28,20,12,4,26,18,10), 
  (1,25,19,13,7)(2,26,20,14,8)(3,27,21,15,9)(4,28,22,16,10)(5,29,23,17,11)(6,30,
    24,18,12), (1,27,23,19,15,11,7,3,29,25,21,17,13,9,5)(2,28,24,20,16,12,8,4,
    30,26,22,18,14,10,6), (1,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(2,30,28,26,
    24,22,20,18,16,14,12,10,8,6,4) ]
gap> Order(Subgroup(c30,[a^4,a^6]));
15
gap> Order(Subgroup(c30,[a^10,a^2]));
15
gap> Order(Subgroup(c30,[a^15,a^2]));
30
gap> Order(Subgroup(c30,[a^9,a^12]));
10
gap> Order(Subgroup(c30,[a^8,a^12]));
15
gap> Order(Subgroup(c30,[a^152]));
15
gap> Order(Subgroup(c30,[a^15]));
2
gap> Order(Subgroup(c30,[a^2]));
15
gap> HELP("Intersection");
gap> 1
> HELP("1");
Syntax error: ; expected
HELP("1");
   ^
"1"
gap> HELP("Intersection");
gap> HELP("1");
gap> 
gap> Intersection(Subgroup(c30,[a^4]),Subgroup(c30,[a^6]));
Group([ (1,7,13,19,25)(2,8,14,20,26)(3,9,15,21,27)(4,10,16,22,28)(5,11,17,23,
    29)(6,12,18,24,30) ])
gap> Order(Intersection(Subgroup(c30,[a^4]),Subgroup(c30,[a^6])));
5
gap> Order(Intersection(Subgroup(c30,[a^10]),Subgroup(c30,[a^2])));
3
gap> Order(Intersection(Subgroup(c30,[a^15]),Subgroup(c30,[a^2])));
1
gap> Order(Subgroup(c30,[a^12]));
5
gap> Read("orderFrequency");
gap> orderFrequency(c6);
[Order of element, Number of that order]=[ [ 1, 1 ], [ 2, 1 ], [ 3, 2 ], 
  [ 6, 2 ] ]
gap> orderFrequency(c30);
[Order of element, Number of that order]=[ [ 1, 1 ], [ 2, 1 ], [ 3, 2 ], 
  [ 5, 4 ], [ 6, 2 ], [ 10, 4 ], [ 15, 8 ], [ 30, 8 ] ]
gap> quit;