PREP - Professional Enhancement Programs of the MAA

Incorporating the Software GAP into Teaching Abstract Algebra

A PREP Workshop

GAP log - Monday July 10, 10 am - 11:30 am

Tutorial

gap> 7+8;
15
gap> 7*9;
63
gap> Gcd(123, 456);
3
gap> Gcdex(4,15);
rec( gcd := 1, coeff1 := 4, coeff2 := -1, coeff3 := -15, coeff4 := 4 )
gap> gcdlist:= Gcdex(4,15);
rec( gcd := 1, coeff1 := 4, coeff2 := -1, coeff3 := -15, coeff4 := 4 )
gap> gcdlist.coeff1;
4
gap> gcdlist.coeff1 +10;
14
gap> gcdlist.coeff1*4 + gcdlist.coeff2*15;
1
gap> square:= x -> x^2;
function( x ) ... end
gap> square(4);
16
gap> SumFirstnInt:= x -> x*(x+1)/2;
function( x ) ... end
gap> SumFirstnInt(100);
5050

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> Factorization(K,(2,3,4));
x1^-1*x2*x1
gap> Factorization(K,(4,5,6));
x1*x2^-1*x1^-1*x2^-1*x1
gap> (1,3)*(2,3);
(1,2,3)
gap> (2,3)*(1,3);
(1,3,2)
gap> r:= (1,2,3,4);
(1,2,3,4)
gap> s:= (1,4,5,6);
(1,4,5,6)
gap> K:= Subgroup(G,[r,s]);
Group([ (1,2,3,4), (1,4,5,6) ])
gap> a:= (1,5,10);
(1,5,10)
gap> a^2;
(1,10,5)
gap> a^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> a*b;
(1,7,9,11)(3,5,10)
gap> CycleStructurePerm((1,2,3)(4,5,6)(7,8)(9,11));
[ 2, 2 ]
gap> CycleStructurePerm((1,2,3));
[ , 1 ]
gap> CycleStructurePerm((1,2,3,4)(3,4));
Permutation: cycles must be disjoint and duplicate-free
gap> CycleStructurePerm((1,2,3,4)(5,6));
[ 1,, 1 ]
gap> cstruc:= function(G,s)
> return Filtered(Elements(G), x -> CycleStructurePerm(x) = s);
> end;
function( G, s ) ... end
gap> cstruc(SymmetricGroup(6), [1,,1]);
[ (1,2)(3,4,5,6), (1,2)(3,4,6,5), (1,2)(3,5,6,4), (1,2)(3,5,4,6), (1,2)(3,6,5,4), 
  (1,2)(3,6,4,5), (1,2,3,4)(5,6), (1,2,3,5)(4,6), (1,2,3,6)(4,5), (1,2,4,3)(5,6), 
  (1,2,4,6)(3,5), (1,2,4,5)(3,6), (1,2,5,3)(4,6), (1,2,5,6)(3,4), (1,2,5,4)(3,6), 
  (1,2,6,3)(4,5), (1,2,6,5)(3,4), (1,2,6,4)(3,5), (1,3,4,2)(5,6), (1,3,5,2)(4,6), 
  (1,3,6,2)(4,5), (1,3)(2,4,5,6), (1,3)(2,4,6,5), (1,3,2,4)(5,6), (1,3,5,6)(2,4), 
  (1,3,6,5)(2,4), (1,3)(2,5,6,4), (1,3)(2,5,4,6), (1,3,2,5)(4,6), (1,3,4,6)(2,5), 
  (1,3,6,4)(2,5), (1,3)(2,6,5,4), (1,3)(2,6,4,5), (1,3,2,6)(4,5), (1,3,4,5)(2,6), 
  (1,3,5,4)(2,6), (1,4,3,2)(5,6), (1,4,6,2)(3,5), (1,4,5,2)(3,6), (1,4,2,3)(5,6), 
  (1,4,5,6)(2,3), (1,4,6,5)(2,3), (1,4)(2,3,5,6), (1,4)(2,3,6,5), (1,4,6,3)(2,5), 
  (1,4)(2,5,6,3), (1,4)(2,5,3,6), (1,4,2,5)(3,6), (1,4,3,6)(2,5), (1,4,5,3)(2,6), 
  (1,4)(2,6,5,3), (1,4)(2,6,3,5), (1,4,2,6)(3,5), (1,4,3,5)(2,6), (1,5,3,2)(4,6), 
  (1,5,6,2)(3,4), (1,5,4,2)(3,6), (1,5,2,3)(4,6), (1,5,6,4)(2,3), (1,5,4,6)(2,3), 
  (1,5)(2,3,4,6), (1,5)(2,3,6,4), (1,5,6,3)(2,4), (1,5)(2,4,6,3), (1,5,2,4)(3,6), 
  (1,5,3,6)(2,4), (1,5)(2,4,3,6), (1,5,4,3)(2,6), (1,5)(2,6,4,3), (1,5,3,4)(2,6), 
  (1,5)(2,6,3,4), (1,5,2,6)(3,4), (1,6,3,2)(4,5), (1,6,5,2)(3,4), (1,6,4,2)(3,5), 
  (1,6,2,3)(4,5), (1,6,5,4)(2,3), (1,6,4,5)(2,3), (1,6)(2,3,4,5), (1,6)(2,3,5,4), 
  (1,6,5,3)(2,4), (1,6)(2,4,5,3), (1,6,2,4)(3,5), (1,6,3,5)(2,4), (1,6)(2,4,3,5), 
  (1,6,4,3)(2,5), (1,6)(2,5,4,3), (1,6,3,4)(2,5), (1,6)(2,5,3,4), (1,6,2,5)(3,4) ]
gap> Size(cstruc(SymmetricGroup(9), [2,,1]));
11340
gap> Size(cstruc(SymmetricGroup(9), [,,1,1]));
18144
gap> Size(Centralizer(SymmetricGroup(9), (1,2,3,4)(5,6)(7,8)));
32
gap> Size(Centralizer(SymmetricGroup(9), (5,1,3,4)(2,6)(7,8)));
32
gap> Factorial(9);
362880
gap> LogTo();

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This PREP workshop is made possible by the NSF grant DUE: 0341481

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