Section 1: Working with Permutation Groups

gap> G:= SymmetricGroup(6);
Sym( [ 1 .. 6 ] )
gap> r:= (1,3,4,5,6);
(1,3,4,5,6)
gap> s:= (1,3,2);
(1,3,2)
gap> K:= Subgroup(G, [r,s]);
Group([ (1,3,4,5,6), (1,3,2) ])
gap> Elements(K);
[ (), (4,5,6), (4,6,5), (3,4)(5,6), (3,4,5), (3,4,6),
(3,5,4), (3,5,6),
(3,5)(4,6), (3,6,4), (3,6,5), (3,6)(4,5),
  (2,3)(5,6), (2,3)(4,5), (2,3)(4,6), (2,3,4), (2,3,4,5,6),
(2,3,4,6,5), (2,3,5,6,4), (2,3,5), (2,3,5,4,6),
  (2,3,6,5,4), (2,3,6), (2,3,6,4,5), (2,4,3), (2,4,5,6,3),
(2,4,6,5,3),
(2,4)(5,6), (2,4,5), (2,4,6), (2,4)(3,5),
  (2,4,3,5,6), (2,4,6,3,5), (2,4)(3,6), (2,4,3,6,5),
(2,4,5,3,6),
(2,5,6,4,3), (2,5,3), (2,5,4,6,3), (2,5,4),
  (2,5,6), (2,5)(4,6), (2,5,6,3,4), (2,5)(3,4), (2,5,3,4,6),
(2,5,3,6,4), (2,5,4,3,6), (2,5)(3,6), (2,6,5,4,3),
  (2,6,3), (2,6,4,5,3), (2,6,4), (2,6,5), (2,6)(4,5),
(2,6,5,3,4),
(2,6)(3,4), (2,6,3,4,5), (2,6,3,5,4),
  (2,6,4,3,5), (2,6)(3,5), (1,2)(5,6), (1,2)(4,5),
(1,2)(4,6),
(1,2)(3,4), (1,2)(3,4,5,6), (1,2)(3,4,6,5),
  (1,2)(3,5,6,4), (1,2)(3,5), (1,2)(3,5,4,6),
(1,2)(3,6,5,4),
(1,2)(3,6), (1,2)(3,6,4,5), (1,2,3), (1,2,3)(4,5,6),
  (1,2,3)(4,6,5), (1,2,3,4)(5,6), (1,2,3,4,5), (1,2,3,4,6),
(1,2,3,5,4), (1,2,3,5,6), (1,2,3,5)(4,6), (1,2,3,6,4),
  (1,2,3,6,5), (1,2,3,6)(4,5), (1,2,4,3)(5,6), (1,2,4,5,3),
(1,2,4,6,3), (1,2,4), (1,2,4,5,6), (1,2,4,6,5),
  (1,2,4)(3,5,6), (1,2,4,3,5), (1,2,4,6)(3,5),
(1,2,4)(3,6,5),
(1,2,4,3,6), (1,2,4,5)(3,6), (1,2,5,4,3),
  (1,2,5,6,3), (1,2,5,3)(4,6), (1,2,5,6,4), (1,2,5),
(1,2,5,4,6),
(1,2,5,3,4), (1,2,5,6)(3,4), (1,2,5)(3,4,6),
  (1,2,5,4)(3,6), (1,2,5)(3,6,4), (1,2,5,3,6), (1,2,6,4,3),
(1,2,6,5,3), (1,2,6,3)(4,5), (1,2,6,5,4), (1,2,6),
  (1,2,6,4,5), (1,2,6,3,4), (1,2,6,5)(3,4), (1,2,6)(3,4,5),
(1,2,6,4)(3,5), (1,2,6)(3,5,4), (1,2,6,3,5), (1,3,2),
  (1,3,2)(4,5,6), (1,3,2)(4,6,5), (1,3,4,2)(5,6),
(1,3,4,5,2),
(1,3,4,6,2), (1,3,5,4,2), (1,3,5,6,2),
  (1,3,5,2)(4,6), (1,3,6,4,2), (1,3,6,5,2), (1,3,6,2)(4,5),
(1,3)(5,6), (1,3)(4,5), (1,3)(4,6), (1,3,4),
  (1,3,4,5,6), (1,3,4,6,5), (1,3,5,6,4), (1,3,5),
(1,3,5,4,6),
(1,3,6,5,4), (1,3,6), (1,3,6,4,5), (1,3)(2,4),
  (1,3)(2,4,5,6), (1,3)(2,4,6,5), (1,3,2,4)(5,6),
(1,3,2,4,5),
(1,3,2,4,6), (1,3,5,2,4), (1,3,5,6)(2,4),
  (1,3,5)(2,4,6), (1,3,6,2,4), (1,3,6,5)(2,4),
(1,3,6)(2,4,5),
(1,3)(2,5,6,4), (1,3)(2,5), (1,3)(2,5,4,6),
  (1,3,2,5,4), (1,3,2,5,6), (1,3,2,5)(4,6), (1,3,4)(2,5,6),
(1,3,4,2,5), (1,3,4,6)(2,5), (1,3,6,4)(2,5),
  (1,3,6)(2,5,4), (1,3,6,2,5), (1,3)(2,6,5,4), (1,3)(2,6),
(1,3)(2,6,4,5), (1,3,2,6,4), (1,3,2,6,5), (1,3,2,6)(4,5),
  (1,3,4)(2,6,5), (1,3,4,2,6), (1,3,4,5)(2,6),
(1,3,5,4)(2,6),
(1,3,5)(2,6,4), (1,3,5,2,6), (1,4,3,2)(5,6),
  (1,4,5,3,2), (1,4,6,3,2), (1,4,2), (1,4,5,6,2),
(1,4,6,5,2),
(1,4,2)(3,5,6), (1,4,3,5,2), (1,4,6,2)(3,5),
  (1,4,2)(3,6,5), (1,4,3,6,2), (1,4,5,2)(3,6), (1,4,3),
(1,4,5,6,3),
(1,4,6,5,3), (1,4)(5,6), (1,4,5), (1,4,6),
  (1,4)(3,5), (1,4,3,5,6), (1,4,6,3,5), (1,4)(3,6),
(1,4,3,6,5),
(1,4,5,3,6), (1,4,2,3)(5,6), (1,4,5,2,3),
  (1,4,6,2,3), (1,4)(2,3), (1,4,5,6)(2,3), (1,4,6,5)(2,3),
(1,4)(2,3,5,6), (1,4,2,3,5), (1,4,6)(2,3,5),
  (1,4)(2,3,6,5), (1,4,2,3,6), (1,4,5)(2,3,6), (1,4,2,5,3),
(1,4,3)(2,5,6), (1,4,6,3)(2,5), (1,4)(2,5,6,3),
  (1,4,3,2,5), (1,4,6)(2,5,3), (1,4)(2,5), (1,4,2,5,6),
(1,4,6,2,5),
(1,4)(2,5,3,6), (1,4,2,5)(3,6), (1,4,3,6)(2,5),
  (1,4,2,6,3), (1,4,3)(2,6,5), (1,4,5,3)(2,6),
(1,4)(2,6,5,3),
(1,4,3,2,6), (1,4,5)(2,6,3), (1,4)(2,6), (1,4,2,6,5),
  (1,4,5,2,6), (1,4)(2,6,3,5), (1,4,2,6)(3,5),
(1,4,3,5)(2,6),
(1,5,4,3,2), (1,5,6,3,2), (1,5,3,2)(4,6),
  (1,5,6,4,2), (1,5,2), (1,5,4,6,2), (1,5,3,4,2),
(1,5,6,2)(3,4),
(1,5,2)(3,4,6), (1,5,4,2)(3,6), (1,5,2)(3,6,4),
  (1,5,3,6,2), (1,5,6,4,3), (1,5,3), (1,5,4,6,3), (1,5,4),
(1,5,6),
(1,5)(4,6), (1,5,6,3,4), (1,5)(3,4),
  (1,5,3,4,6), (1,5,3,6,4), (1,5,4,3,6), (1,5)(3,6),
(1,5,4,2,3),
(1,5,6,2,3), (1,5,2,3)(4,6), (1,5,6,4)(2,3),
  (1,5)(2,3), (1,5,4,6)(2,3), (1,5,2,3,4), (1,5,6)(2,3,4),
(1,5)(2,3,4,6), (1,5,4)(2,3,6), (1,5)(2,3,6,4),
  (1,5,2,3,6), (1,5,6,3)(2,4), (1,5,2,4,3), (1,5,3)(2,4,6),
(1,5,3,2,4), (1,5,6)(2,4,3), (1,5)(2,4,6,3),
  (1,5,6,2,4), (1,5)(2,4), (1,5,2,4,6), (1,5,2,4)(3,6),
(1,5,3,6)(2,4), (1,5)(2,4,3,6), (1,5,3)(2,6,4),
  (1,5,4,3)(2,6), (1,5,2,6,3), (1,5,4)(2,6,3),
(1,5)(2,6,4,3),
(1,5,3,2,6), (1,5,2,6,4), (1,5,4,2,6), (1,5)(2,6),
  (1,5,3,4)(2,6), (1,5)(2,6,3,4), (1,5,2,6)(3,4),
(1,6,4,3,2),
(1,6,5,3,2), (1,6,3,2)(4,5), (1,6,5,4,2), (1,6,2),
  (1,6,4,5,2), (1,6,3,4,2), (1,6,5,2)(3,4), (1,6,2)(3,4,5),
(1,6,4,2)(3,5), (1,6,2)(3,5,4), (1,6,3,5,2),
  (1,6,5,4,3), (1,6,3), (1,6,4,5,3), (1,6,4), (1,6,5),
(1,6)(4,5),
(1,6,5,3,4), (1,6)(3,4), (1,6,3,4,5),
  (1,6,3,5,4), (1,6,4,3,5), (1,6)(3,5), (1,6,4,2,3),
(1,6,5,2,3),
(1,6,2,3)(4,5), (1,6,5,4)(2,3), (1,6)(2,3),
  (1,6,4,5)(2,3), (1,6,2,3,4), (1,6,5)(2,3,4),
(1,6)(2,3,4,5),
(1,6,4)(2,3,5), (1,6)(2,3,5,4), (1,6,2,3,5),
  (1,6,5,3)(2,4), (1,6,2,4,3), (1,6,3)(2,4,5), (1,6,3,2,4),
(1,6,5)(2,4,3), (1,6)(2,4,5,3), (1,6,5,2,4), (1,6)(2,4),
  (1,6,2,4,5), (1,6,2,4)(3,5), (1,6,3,5)(2,4),
(1,6)(2,4,3,5),
(1,6,3)(2,5,4), (1,6,4,3)(2,5), (1,6,2,5,3),
  (1,6,4)(2,5,3), (1,6)(2,5,4,3), (1,6,3,2,5), (1,6,2,5,4),
(1,6,4,2,5), (1,6)(2,5), (1,6,3,4)(2,5), (1,6)(2,5,3,4),
  (1,6,2,5)(3,4) ]
gap> Size(K);
360
gap> Factorization(K,(2,3,4));
x1^-1*x2*x1

Exercise 1.1
gap> Factorization(K,(4,5,6));
x1*x2^-1*x1^-1*x2^-1*x1
gap> Factorization(K,(2,3));
fail

Exercise 1.2
gap> r:= (1,4,3,2);
(1,4,3,2)
gap> s:=(1,4,5,6);
(1,4,5,6)
gap> K:= Subgroup(G, [r,s]);
Group([ (1,4,3,2), (1,4,5,6) ])
gap> Size(K);
120
gap> Factorization(K,(2,3));
fail
gap> Factorization(K,(1,2)(3,4));
fail

Exercise 1.3
gap> b:= (1,3,5,7,9,11);
(1,3,5,7,9,11)
gap> b^2;
(1,5,9)(3,7,11)
gap> b^3;
(1,7)(3,9)(5,11)
gap> b^6;
()
gap> a:= (1,5,10);
(1,5,10)
gap> a*b;
(1,7,9,11)(3,5,10)
gap> Order(a*b);
12
gap> (a*b)^3;
(1,11,9,7)

gap> CycleStructurePerm((1,2,3)(4,6,8,9,10,11));
[ , 1,,, 1 ]
gap> CycleStructurePerm((1,2)(4,6,8,9));
[ 1,, 1 ]
gap> cstruc:= function(G,s)
> return Filtered(Elements(G), x -> CycleStructurePerm(x) =
s);
> end;
function( G, s ) ... end
gap> cstruc(SymmetricGroup(7),[1,1]);
[ (3,4)(5,6,7), (3,4)(5,7,6), (3,4,5)(6,7), (3,4,6)(5,7),
(3,4,7)(5,6), (3,5,4)(6,7), (3,5)(4,6,7), (3,5,7)(4,6),
  (3,5)(4,7,6), (3,5,6)(4,7), (3,6,4)(5,7), (3,6,7)(4,5),
(3,6)(4,5,7), (3,6,5)(4,7), (3,6)(4,7,5), (3,7,4)(5,6),
  (3,7,6)(4,5), (3,7)(4,5,6), (3,7,5)(4,6), (3,7)(4,6,5),
(2,3)(5,6,7), (2,3)(5,7,6), (2,3)(4,5,6), (2,3)(4,5,7),
  (2,3)(4,6,5), (2,3)(4,6,7), (2,3)(4,7,5), (2,3)(4,7,6),
(2,3,4)(6,7), (2,3,4)(5,6), (2,3,4)(5,7), (2,3,5)(6,7),
  (2,3,5)(4,6), (2,3,5)(4,7), (2,3,6)(5,7), (2,3,6)(4,5),
(2,3,6)(4,7), (2,3,7)(5,6), (2,3,7)(4,5), (2,3,7)(4,6),
  (2,4,3)(6,7), (2,4,3)(5,6), (2,4,3)(5,7), (2,4)(5,6,7),
(2,4)(5,7,6), (2,4,5)(6,7), (2,4,6)(5,7), (2,4,7)(5,6),
  (2,4)(3,5,6), (2,4)(3,5,7), (2,4,6)(3,5), (2,4,7)(3,5),
(2,4)(3,6,5), (2,4)(3,6,7), (2,4,5)(3,6), (2,4,7)(3,6),
  (2,4)(3,7,5), (2,4)(3,7,6), (2,4,5)(3,7), (2,4,6)(3,7),
(2,5,3)(6,7), (2,5,3)(4,6), (2,5,3)(4,7), (2,5,4)(6,7),
  (2,5)(4,6,7), (2,5,7)(4,6), (2,5)(4,7,6), (2,5,6)(4,7),
(2,5,6)(3,4), (2,5,7)(3,4), (2,5)(3,4,6), (2,5)(3,4,7),
  (2,5,4)(3,6), (2,5)(3,6,4), (2,5)(3,6,7), (2,5,7)(3,6),
(2,5,4)(3,7), (2,5)(3,7,4), (2,5)(3,7,6), (2,5,6)(3,7),
  (2,6,3)(5,7), (2,6,3)(4,5), (2,6,3)(4,7), (2,6,4)(5,7),
(2,6,7)(4,5), (2,6)(4,5,7), (2,6,5)(4,7), (2,6)(4,7,5),
  (2,6,5)(3,4), (2,6,7)(3,4), (2,6)(3,4,5), (2,6)(3,4,7),
(2,6,4)(3,5), (2,6)(3,5,4), (2,6,7)(3,5), (2,6)(3,5,7),
  (2,6,4)(3,7), (2,6)(3,7,4), (2,6,5)(3,7), (2,6)(3,7,5),
(2,7,3)(5,6), (2,7,3)(4,5), (2,7,3)(4,6), (2,7,4)(5,6),
  (2,7,6)(4,5), (2,7)(4,5,6), (2,7,5)(4,6), (2,7)(4,6,5),
(2,7,5)(3,4), (2,7,6)(3,4), (2,7)(3,4,5), (2,7)(3,4,6),
  (2,7,4)(3,5), (2,7)(3,5,4), (2,7,6)(3,5), (2,7)(3,5,6),
(2,7,4)(3,6), (2,7)(3,6,4), (2,7,5)(3,6), (2,7)(3,6,5),
  (1,2)(5,6,7), (1,2)(5,7,6), (1,2)(4,5,6), (1,2)(4,5,7),
(1,2)(4,6,5), (1,2)(4,6,7), (1,2)(4,7,5), (1,2)(4,7,6),
  (1,2)(3,4,5), (1,2)(3,4,6), (1,2)(3,4,7), (1,2)(3,5,4),
(1,2)(3,5,6), (1,2)(3,5,7), (1,2)(3,6,4), (1,2)(3,6,5),
  (1,2)(3,6,7), (1,2)(3,7,4), (1,2)(3,7,5), (1,2)(3,7,6),
(1,2,3)(6,7), (1,2,3)(5,6), (1,2,3)(5,7), (1,2,3)(4,5),
  (1,2,3)(4,6), (1,2,3)(4,7), (1,2,4)(6,7), (1,2,4)(5,6),
(1,2,4)(5,7), (1,2,4)(3,5), (1,2,4)(3,6), (1,2,4)(3,7),
  (1,2,5)(6,7), (1,2,5)(4,6), (1,2,5)(4,7), (1,2,5)(3,4),
(1,2,5)(3,6), (1,2,5)(3,7), (1,2,6)(5,7), (1,2,6)(4,5),
  (1,2,6)(4,7), (1,2,6)(3,4), (1,2,6)(3,5), (1,2,6)(3,7),
(1,2,7)(5,6), (1,2,7)(4,5), (1,2,7)(4,6), (1,2,7)(3,4),
  (1,2,7)(3,5), (1,2,7)(3,6), (1,3,2)(6,7), (1,3,2)(5,6),
(1,3,2)(5,7), (1,3,2)(4,5), (1,3,2)(4,6), (1,3,2)(4,7),
  (1,3)(5,6,7), (1,3)(5,7,6), (1,3)(4,5,6), (1,3)(4,5,7),
(1,3)(4,6,5), (1,3)(4,6,7), (1,3)(4,7,5), (1,3)(4,7,6),
  (1,3,4)(6,7), (1,3,4)(5,6), (1,3,4)(5,7), (1,3,5)(6,7),
(1,3,5)(4,6), (1,3,5)(4,7), (1,3,6)(5,7), (1,3,6)(4,5),
  (1,3,6)(4,7), (1,3,7)(5,6), (1,3,7)(4,5), (1,3,7)(4,6),
(1,3)(2,4,5), (1,3)(2,4,6), (1,3)(2,4,7), (1,3,5)(2,4),
  (1,3,6)(2,4), (1,3,7)(2,4), (1,3)(2,5,4), (1,3)(2,5,6),
(1,3)(2,5,7), (1,3,4)(2,5), (1,3,6)(2,5), (1,3,7)(2,5),
  (1,3)(2,6,4), (1,3)(2,6,5), (1,3)(2,6,7), (1,3,4)(2,6),
(1,3,5)(2,6), (1,3,7)(2,6), (1,3)(2,7,4), (1,3)(2,7,5),
  (1,3)(2,7,6), (1,3,4)(2,7), (1,3,5)(2,7), (1,3,6)(2,7),
(1,4,2)(6,7), (1,4,2)(5,6), (1,4,2)(5,7), (1,4,2)(3,5),
  (1,4,2)(3,6), (1,4,2)(3,7), (1,4,3)(6,7), (1,4,3)(5,6),
(1,4,3)(5,7), (1,4)(5,6,7), (1,4)(5,7,6), (1,4,5)(6,7),
  (1,4,6)(5,7), (1,4,7)(5,6), (1,4)(3,5,6), (1,4)(3,5,7),
(1,4,6)(3,5), (1,4,7)(3,5), (1,4)(3,6,5), (1,4)(3,6,7),
  (1,4,5)(3,6), (1,4,7)(3,6), (1,4)(3,7,5), (1,4)(3,7,6),
(1,4,5)(3,7), (1,4,6)(3,7), (1,4,5)(2,3), (1,4,6)(2,3),
  (1,4,7)(2,3), (1,4)(2,3,5), (1,4)(2,3,6), (1,4)(2,3,7),
(1,4,3)(2,5), (1,4)(2,5,3), (1,4)(2,5,6), (1,4)(2,5,7),
  (1,4,6)(2,5), (1,4,7)(2,5), (1,4,3)(2,6), (1,4)(2,6,3),
(1,4)(2,6,5), (1,4)(2,6,7), (1,4,5)(2,6), (1,4,7)(2,6),
  (1,4,3)(2,7), (1,4)(2,7,3), (1,4)(2,7,5), (1,4)(2,7,6),
(1,4,5)(2,7), (1,4,6)(2,7), (1,5,2)(6,7), (1,5,2)(4,6),
  (1,5,2)(4,7), (1,5,2)(3,4), (1,5,2)(3,6), (1,5,2)(3,7),
(1,5,3)(6,7), (1,5,3)(4,6), (1,5,3)(4,7), (1,5,4)(6,7),
  (1,5)(4,6,7), (1,5,7)(4,6), (1,5)(4,7,6), (1,5,6)(4,7),
(1,5,6)(3,4), (1,5,7)(3,4), (1,5)(3,4,6), (1,5)(3,4,7),
  (1,5,4)(3,6), (1,5)(3,6,4), (1,5)(3,6,7), (1,5,7)(3,6),
(1,5,4)(3,7), (1,5)(3,7,4), (1,5)(3,7,6), (1,5,6)(3,7),
  (1,5,4)(2,3), (1,5,6)(2,3), (1,5,7)(2,3), (1,5)(2,3,4),
(1,5)(2,3,6), (1,5)(2,3,7), (1,5,3)(2,4), (1,5)(2,4,3),
  (1,5,6)(2,4), (1,5,7)(2,4), (1,5)(2,4,6), (1,5)(2,4,7),
(1,5,3)(2,6), (1,5)(2,6,3), (1,5,4)(2,6), (1,5)(2,6,4),
  (1,5)(2,6,7), (1,5,7)(2,6), (1,5,3)(2,7), (1,5)(2,7,3),
(1,5,4)(2,7), (1,5)(2,7,4), (1,5)(2,7,6), (1,5,6)(2,7),
  (1,6,2)(5,7), (1,6,2)(4,5), (1,6,2)(4,7), (1,6,2)(3,4),
(1,6,2)(3,5), (1,6,2)(3,7), (1,6,3)(5,7), (1,6,3)(4,5),
  (1,6,3)(4,7), (1,6,4)(5,7), (1,6,7)(4,5), (1,6)(4,5,7),
(1,6,5)(4,7), (1,6)(4,7,5), (1,6,5)(3,4), (1,6,7)(3,4),
  (1,6)(3,4,5), (1,6)(3,4,7), (1,6,4)(3,5), (1,6)(3,5,4),
(1,6,7)(3,5), (1,6)(3,5,7), (1,6,4)(3,7), (1,6)(3,7,4),
  (1,6,5)(3,7), (1,6)(3,7,5), (1,6,4)(2,3), (1,6,5)(2,3),
(1,6,7)(2,3), (1,6)(2,3,4), (1,6)(2,3,5), (1,6)(2,3,7),
  (1,6,3)(2,4), (1,6)(2,4,3), (1,6,5)(2,4), (1,6,7)(2,4),
(1,6)(2,4,5), (1,6)(2,4,7), (1,6,3)(2,5), (1,6)(2,5,3),
  (1,6,4)(2,5), (1,6)(2,5,4), (1,6,7)(2,5), (1,6)(2,5,7),
(1,6,3)(2,7), (1,6)(2,7,3), (1,6,4)(2,7), (1,6)(2,7,4),
  (1,6,5)(2,7), (1,6)(2,7,5), (1,7,2)(5,6), (1,7,2)(4,5),
(1,7,2)(4,6), (1,7,2)(3,4), (1,7,2)(3,5), (1,7,2)(3,6),
  (1,7,3)(5,6), (1,7,3)(4,5), (1,7,3)(4,6), (1,7,4)(5,6),
(1,7,6)(4,5), (1,7)(4,5,6), (1,7,5)(4,6), (1,7)(4,6,5),
  (1,7,5)(3,4), (1,7,6)(3,4), (1,7)(3,4,5), (1,7)(3,4,6),
(1,7,4)(3,5), (1,7)(3,5,4), (1,7,6)(3,5), (1,7)(3,5,6),
  (1,7,4)(3,6), (1,7)(3,6,4), (1,7,5)(3,6), (1,7)(3,6,5),
(1,7,4)(2,3), (1,7,5)(2,3), (1,7,6)(2,3), (1,7)(2,3,4),
  (1,7)(2,3,5), (1,7)(2,3,6), (1,7,3)(2,4), (1,7)(2,4,3),
(1,7,5)(2,4), (1,7,6)(2,4), (1,7)(2,4,5), (1,7)(2,4,6),
  (1,7,3)(2,5), (1,7)(2,5,3), (1,7,4)(2,5), (1,7)(2,5,4),
(1,7,6)(2,5), (1,7)(2,5,6), (1,7,3)(2,6), (1,7)(2,6,3),
  (1,7,4)(2,6), (1,7)(2,6,4), (1,7,5)(2,6), (1,7)(2,6,5) ]
gap> Size(cstruc(SymmetricGroup(7),[1,1]));
420

Exercise 1.4

gap> Size(cstruc(SymmetricGroup(9),[2,,1]));
11340
gap> Size(cstruc(SymmetricGroup(9),[,,1,1]));
18144
gap> Size(cstruc(SymmetricGroup(9),[,3]));
2240
gap> Size(cstruc(SymmetricGroup(9),[4]));
945
gap> Centralizer(SymmetricGroup(4),(1,2));
Group([ (1,2), (3,4) ])

Exercise 1.5

gap> G:= SymmetricGroup(9);
Sym( [ 1 .. 9 ] )
gap> Size(Centralizer(G, (1,2,3,4)(5,6)(7,8)));
32
gap> Size(Centralizer(G, (5,1,3,4)(2,6)(7,8)));
32
gap> Size(Centralizer(G, (1,2,3,4,5)(6,7,8,9)));
20
gap> Size(Centralizer(G, (1,2,3)(4,5,6)(7,8,9)));
162
gap> Size(G);
362880

Exercise 1.7
gap> a:= (1,2,3)(5,7,9);
(1,2,3)(5,7,9)
gap> b1:= (1,2,3,4,5,6,7,8,9);
(1,2,3,4,5,6,7,8,9)
gap> b*a*b^-1;
(1,11,2)(3,5,7)
gap> b2:= (1,2)(3,4)(5,6)(7,8)
> ;
(1,2)(3,4)(5,6)(7,8)
gap> b1*a*b1^-1;
(1,2,9)(4,6,8)
gap> b2*a*b2^-1;
(1,4,2)(6,8,9)

Exercise 1.11
gap> Order(G);
362880
gap> Order((1,2));
2
gap> Order((1,3,4)(5,6,8));
3
gap> a*b1;
(1,3,2,4,5,8,9,6,7)
gap> Order(a*b1);
9
gap> Order(b1*a);
9
gap> Order(b2*a);
15
gap> Order(a*b2);
15

Exercise 1.13

gap> H:= Group([(1,2), (1,2,3)]);
Group([ (1,2), (1,2,3) ])
gap> Size(H);
6
gap> Size(SymmetricGroup(3));
6
gap> H:= Group([(1,2), (1,2,3,4)]);
Group([ (1,2), (1,2,3,4) ])
gap> Size(H);
24
gap> Size(SymmetricGroup(4));
24
gap> Size(Group([(1,2), (1,2,3,4,5)]));
120
gap> Size(SymmetricGroup(5));
120
gap> Size(Group([(1,2), (1,2,3,4,5,6)]));
720
gap> Size(SymmetricGroup(6));
720

Exercise 1.14
gap> Size(Group([(1,3,2), (1,3,4)]));
12
gap> Size(SymmetricGroup(4));
24
gap> Size(Group([(1,3,2), (1,3,4,5)]));
120
gap> Size(SymmetricGroup(5));
120
gap> Size(Group([(1,3,2), (1,3,4,5,6)]));
360
gap> Size(SymmetricGroup(6));
720
gap> Size(Group([(1,3,2), (1,3,4,5,6,7)]));
5040
gap> Size(SymmetricGroup(7));
5040
gap>