PREP - Professional Enhancement Programs of the MAA

Exploring Abstract Algebra with Computer Software

A PREP Workshop

GAP log - Tuesday June 29, 2:30 pm - 4:00 pm

gap> G:= DihedralGroup(IsPermGroup, 16);;
gap> obj:= [G];
[ Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ]) ]
gap> code:= ApplicableMethod(IsCyclic, obj,1);
#I  Searching Method for IsCyclic with 1 arguments:
#I  Total: 6 entries
#I  Method 3: ``IsCyclic'', value: 22
function( G ) ... end
gap> print(code);
Variable: 'print' must have a value

gap> Print(code);
function ( G )
    if Length( GeneratorsOfGroup( G ) ) = 1  then
        return true;
    else
        return TRY_NEXT_METHOD;
    fi;
    return;
endgap>

Section 6: Factor Groups

gap> S6 := SymmetricGroup(6);
Sym( [ 1 .. 6 ] )
gap> A6 := AlternatingGroup(6);
Alt( [ 1 .. 6 ] )
gap> D6 := DihedralGroup(IsPermGroup, 12);
Group([ (1,2,3,4,5,6), (2,6)(3,5) ])
gap> Z6 := Center(D6);
Group([ (1,4)(2,5)(3,6) ])
gap> IsNormal(S6,A6);
true
gap> IsNormal(S6,D6);
false
gap> IsNormal(D6,Z6);
true
gap> RightCosets(S6,A6);
[ RightCoset(AlternatingGroup( [ 1 .. 6 ] ),()), RightCoset(AlternatingGroup( [ 1 .. 6 ] ),(5,6)) ]
gap> Order(FactorGroup(S6,A6));
2
gap> RightCosets(D6,Z6);
[ RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),()), RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(2,6)(3,5)), 
  RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,5,3)(2,6,4)), RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,5)
    (2,4)), RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,3,5)(2,4,6)), 
  RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,3)(4,6)) ]
gap> Order(FactorGroup(D6,Z6));
6
gap> H := FactorGroup(D6,Z6);
Group([ f1, f2 ])
gap> KnownAttributesOfObject(H);
[ "Size", "OneImmutable", "Order", "ParentAttr", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement", 
  "FamilyPcgs", "HomePcgs" ]
gap> Size(RightCosets(D6,Z6));
6
gap> H;
Group([ f1, f2 ])
gap> IsAbelian(H);
false
gap> IsomorphismGroups(H,SymmetricGroup(3));
[ f1, f2 ] -> [ (2,3), (1,2,3) ]
gap> Elements(RightCoset(Z6,(2,6)(3,5)));
[ (2,6)(3,5), (1,4)(2,3)(5,6) ]

Exercise 6.1
gap> D8 := DihedralGroup(IsPermGroup,16);
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
gap> Z8 := Center(D8);
Group([ (1,5)(2,6)(3,7)(4,8) ])
gap> RightCosets(D8,Z8);
[ RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),()), RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),
    (2,8)(3,7)(4,6)), RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,7,5,3)(2,8,6,4)), 
  RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,7)(2,6)(3,5)), 
  RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,8,7,6,5,4,3,2)), 
  RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,8)(2,7)(3,6)(4,5)), 
  RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,6,3,8,5,2,7,4)), 
  RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,6)(2,5)(3,4)(7,8)) ]
gap> e := Elements(SymmetricGroup(3));
[ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]

gap> PrintArray(MultiplicationTable(e));
[ [  1,  2,  3,  4,  5,  6 ],
  [  2,  1,  4,  3,  6,  5 ],
  [  3,  5,  1,  6,  2,  4 ],
  [  4,  6,  2,  5,  1,  3 ],
  [  5,  3,  6,  1,  4,  2 ],
  [  6,  4,  5,  2,  3,  1 ] ]

Exercise 6.3
gap> F := FactorGroup(D8,Z8);
Group([ f1, f2, f3 ])
gap> Elements(F);
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
gap> e2:=Elements(F);
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
gap> PrintArray(MultiplicationTable(e2));
[ [  1,  2,  3,  4,  5,  6,  7,  8 ],
  [  2,  1,  5,  6,  3,  4,  8,  7 ],
  [  3,  8,  4,  7,  2,  5,  1,  6 ],
  [  4,  6,  7,  1,  8,  2,  3,  5 ],
  [  5,  7,  6,  8,  1,  3,  2,  4 ],
  [  6,  4,  8,  2,  7,  1,  5,  3 ],
  [  7,  5,  1,  3,  6,  8,  4,  2 ],
  [  8,  3,  2,  5,  4,  7,  6,  1 ] ]
gap> F := D8/Z8;
Group([ f1, f2, f3 ])
gap> Elements(F);
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
gap> e2 := Elements(F);
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
gap> PrintArray(MultiplicationTable(e2));
[ [  1,  2,  3,  4,  5,  6,  7,  8 ],
  [  2,  1,  5,  6,  3,  4,  8,  7 ],
  [  3,  8,  4,  7,  2,  5,  1,  6 ],
  [  4,  6,  7,  1,  8,  2,  3,  5 ],
  [  5,  7,  6,  8,  1,  3,  2,  4 ],
  [  6,  4,  8,  2,  7,  1,  5,  3 ],
  [  7,  5,  1,  3,  6,  8,  4,  2 ],
  [  8,  3,  2,  5,  4,  7,  6,  1 ] ]
gap> IsomorphismGroups(F,DihedralGroup(IsPermGroup,8));
[ f1, f2, f3 ] -> [ (2,4), (1,2,3,4), (1,3)(2,4) ]

Exercise 6.4
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> Z10 := Center(D10);
Group([ (1,6)(2,7)(3,8)(4,9)(5,10) ])
gap> F := D10/Z10;
Group([ f1, f2 ])
gap> e10 := Elements(F);
[ <identity> of ..., f1, f2, f1*f2, f2^2, f1*f2^2, f2^3, f1*f2^3, f2^4, f1*f2^4 ]
gap> PrintArray(MultiplicationTable(e10));
[ [   1,   2,   3,   4,   5,   6,   7,   8,   9,  10 ],
  [   2,   1,   4,   3,   6,   5,   8,   7,  10,   9 ],
  [   3,  10,   5,   2,   7,   4,   9,   6,   1,   8 ],
  [   4,   9,   6,   1,   8,   3,  10,   5,   2,   7 ],
  [   5,   8,   7,  10,   9,   2,   1,   4,   3,   6 ],
  [   6,   7,   8,   9,  10,   1,   2,   3,   4,   5 ],
  [   7,   6,   9,   8,   1,  10,   3,   2,   5,   4 ],
  [   8,   5,  10,   7,   2,   9,   4,   1,   6,   3 ],
  [   9,   4,   1,   6,   3,   8,   5,  10,   7,   2 ],
  [  10,   3,   2,   5,   4,   7,   6,   9,   8,   1 ] ]
gap> IsomorphismGroups(F,DihedralGroup(IsPermGroup,10));
[ f1, f2 ] -> [ (2,5)(3,4), (1,2,3,4,5) ]
gap> D12 := DihedralGroup(IsPermGroup,24);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12), (2,12)(3,11)(4,10)(5,9)(6,8) ])
gap> Z12 := Center(D12);
Group([ (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) ])
gap> F := D12/Z12;
Group([ f1, f2, f3 ])
gap> e12 := Elements(F);
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f3^2, f1*f2*f3, f1*f3^2, f2*f3^2, f1*f2*f3^2 ]
gap> PrintArray(MultiplicationTable(e12));
[ [   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12 ],
  [   2,   1,   5,   6,   3,   4,   9,  10,   7,   8,  12,  11 ],
  [   3,   9,   8,   7,   2,  12,   1,  11,   6,   5,   4,  10 ],
  [   4,  10,   7,   8,  12,   2,  11,   1,   5,   6,   3,   9 ],
  [   5,   7,  10,   9,   1,  11,   2,  12,   4,   3,   6,   8 ],
  [   6,   8,   9,  10,  11,   1,  12,   2,   3,   4,   5,   7 ],
  [   7,   5,   1,  11,  10,   9,   4,   3,   2,  12,   8,   6 ],
  [   8,   6,  11,   1,   9,  10,   3,   4,  12,   2,   7,   5 ],
  [   9,   3,   2,  12,   8,   7,   6,   5,   1,  11,  10,   4 ],
  [  10,   4,  12,   2,   7,   8,   5,   6,  11,   1,   9,   3 ],
  [  11,  12,   4,   3,   6,   5,   8,   7,  10,   9,   1,   2 ],
  [  12,  11,   6,   5,   4,   3,  10,   9,   8,   7,   2,   1 ] ]
gap> IsomorphismGroups(F,DihedralGroup(IsPermGroup,12));
[ f1, f2, f3 ] -> [ (1,6)(2,5)(3,4), (1,2,3,4,5,6), (1,5,3)(2,6,4) ]

Section 7: Group Homomorphisms

gap> S3:= SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> f1:= GroupHomomorphismByImages(S3, S3, [(1,2,3),(1,3)], [(1,3,2), (1,2)]);
[ (1,2,3), (1,3) ] -> [ (1,3,2), (1,2) ]
gap> Image(f1, (2,3));
(2,3)
gap> Image(f1, (1,2));
(1,3)
gap> f1:= GroupHomomorphismByImages(S3, S3, [(1,2,3),(1,3)], [(1,2), (1,2)]);
fail
gap> f1:= GroupHomomorphismByImages(S3, S3, [(1,2,3),(1,3)], [(1,3,2), (1,2)]);
[ (1,2,3), (1,3) ] -> [ (1,3,2), (1,2) ]
gap> Kernel(f1);
Group(())
gap> Read("autoDn");
gap> Read("homoDn");
gap> d6:= DihedralGroup(IsPermGroup,12);
Group([ (1,2,3,4,5,6), (2,6)(3,5) ])
gap> autoDn(d6);
[ [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,2,3,4,5,6), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,2,3,4,5,6), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,6,5,4,3,2), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,6,5,4,3,2), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,6)(2,5)(3,4) ] ]
gap> homoDn(d6);
[ [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (2,6)(3,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,4)(2,5)(3,6) ],
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (2,6)(3,5), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (2,6)(3,5), (2,6)(3,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (2,6)(3,5), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (2,6)(3,5), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2)(3,6)(4,5), () ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2)(3,6)(4,5), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,2)(3,6)(4,5), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2)(3,6)(4,5), (1,5)(2,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,2,3,4,5,6), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,2,3,4,5,6), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3)(4,6), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3)(4,6), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3)(4,6), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,3)(4,6), (1,6)(2,5)(3,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (2,6)(3,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,3,5)(2,4,6), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (1,4)(2,3)(5,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,3,5)(2,4,6), (1,6)(2,5)(3,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,3)(5,6), () ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,3)(5,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,4)(2,3)(5,6), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,3)(5,6), (1,4)(2,5)(3,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,4)(2,5)(3,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,2)(3,6)(4,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,4)(2,5)(3,6), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,4)(2,5)(3,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,4)(2,5)(3,6), (1,6)(2,5)(3,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5)(2,4), () ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5)(2,4), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,5)(2,4), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5)(2,4), (1,5)(2,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,5,3)(2,6,4), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,5,3)(2,6,4), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,6,5,4,3,2), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,6,5,4,3,2), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6)(2,5)(3,4), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,6)(2,5)(3,4), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6)(2,5)(3,4), (1,4)(2,5)(3,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6)(2,5)(3,4), (1,6)(2,5)(3,4) ] ]

Exercise 7.2
gap> d4:= DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> autoDn(d4);
[ [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,4)(2,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,4)(2,3) ] ]
gap> f1:= GroupHomomorphismByImages(d4, d4, [(1,2,3,4),(1,3)], [(1,2,3,4), (2,4)]);
[ (1,2,3,4), (1,3) ] -> [ (1,2,3,4), (2,4) ]
gap> Image(f1, (2,4));
(1,3)
gap> homoDn(d4);
[ [ (1,2,3,4), (2,4) ] -> [ (), () ], [ (1,2,3,4), (2,4) ] -> [ (), (2,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (), (1,2)(3,4) ], [ (1,2,3,4), (2,4) ] -> [ (), (1,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (), (1,3)(2,4) ], [ (1,2,3,4), (2,4) ] -> [ (), (1,4)(2,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (2,4), () ], [ (1,2,3,4), (2,4) ] -> [ (2,4), (2,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (2,4), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (2,4), (1,3)(2,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,2)(3,4), () ], [ (1,2,3,4), (2,4) ] -> [ (1,2)(3,4), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,2)(3,4), (1,3)(2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,2)(3,4), (1,4)(2,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,4)(2,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,3), () ], [ (1,2,3,4), (2,4) ] -> [ (1,3), (2,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,3), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (1,3), (1,3)(2,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,3)(2,4), () ], [ (1,2,3,4), (2,4) ] -> [ (1,3)(2,4), (2,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,3)(2,4), (1,2)(3,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,3)(2,4), (1,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,3)(2,4), (1,3)(2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,3)(2,4), (1,4)(2,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,4)(2,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4)(2,3), () ], [ (1,2,3,4), (2,4) ] -> [ (1,4)(2,3), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4)(2,3), (1,3)(2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,4)(2,3), (1,4)(2,3) ] ]

Exercise 7.4
gap> d19:= DihedralGroup(IsPermGroup,38);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11) ])
gap> Size(homoDn(d19));
362
gap> Size(autoDn(d19));
342
gap> d21:= DihedralGroup(IsPermGroup,42);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21), (2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,
    13)(11,12) ])
gap> Size(autoDn(d21));
252
gap> Size(homoDn(d21));
442

Section 8: Introduction to Rings

gap> r:= Integers mod 8;
(Integers mod 8)
gap> Elements(r);
[ ZmodnZObj( 0, 8 ), ZmodnZObj( 1, 8 ), ZmodnZObj( 2, 8 ), ZmodnZObj( 3, 8 ), ZmodnZObj( 4, 8 ), ZmodnZObj( 5, 8 ), 
  ZmodnZObj( 6, 8 ), ZmodnZObj( 7, 8 ) ]
gap> s:= Integers mod 7;
GF(7)
gap> Elements(s);
[ 0*Z(7), Z(7)^0, Z(7), Z(7)^2, Z(7)^3, Z(7)^4, Z(7)^5 ]
gap> Int(Z(7))
> ;
3
gap> 5*One(s);
Z(7)^5
gap> 3^5  mod 7;
5

Exercise 8.1
gap> R:= Integers mod 3;
GF(3)
gap> Elements(R);
[ 0*Z(3), Z(3)^0, Z(3) ]
gap> Int(Z(3));
2
gap> z:= Z(3);
Z(3)
gap> z^2+z^2;
Z(3)
gap> (z^0)^2+z^2;
Z(3)
gap> R:= Integers mod 5;
GF(5)
gap> Elements(R);
[ 0*Z(5), Z(5)^0, Z(5), Z(5)^2, Z(5)^3 ]
gap> Int(Z(5))
> ;
2
gap> z:= Z(5);
Z(5)
gap> (z^0)^2+z^2;
0*Z(5)
gap> R:= Integers mod 11;
GF(11)
gap> Elements(R);
[ 0*Z(11), Z(11)^0, Z(11), Z(11)^2, Z(11)^3, Z(11)^4, Z(11)^5, Z(11)^6, Z(11)^7, Z(11)^8, Z(11)^9 ]
gap> Int(Z(11))
> ;
2
gap> z:= Z(11);
Z(11)
gap> (z^0)^2+z^2;
Z(11)^4
gap> (z)^2+z^2;
Z(11)^3
gap> Read("intror");
gap> intror(3);
false
gap> intror(5);
true
gap> intror(7);
false

gap> M:= Integers mod 6;
(Integers mod 6)
gap> Idempotents(M);
[ ZmodnZObj( 0, 6 ), ZmodnZObj( 1, 6 ), ZmodnZObj( 3, 6 ), ZmodnZObj( 4, 6 ) ]
gap> N:= Integers mod 9;
(Integers mod 9)
gap> Idempotents(N);
[ ZmodnZObj( 0, 9 ), ZmodnZObj( 1, 9 ) ]
gap> Read("nilpotentCount");
gap> nilpotentCount(N);
3
gap> nilpotentCount(M);
1
gap> quit;

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