PREP - Professional Enhancement Programs of the MAA

Exploring Abstract Algebra with Computer Software

A PREP Workshop

GAP log - Friday July 2, 2:30 pm - 4:00 pm

Section 16: Gaussian Integers

gap> R:= GaussianIntegers;
GaussianIntegers
gap> i:= E(4);
E(4)
gap> i^2;
-1
gap> Factors(R,4);
[ -1-E(4), 1+E(4), 1+E(4), 1+E(4) ]
gap> Factors(4);
[ 2, 2 ]

Exercise 16.1
gap> Factors(R,2); Factors(R,3); Factors(R,5);
[ 1-E(4), 1+E(4) ]
[ 3 ]
[ 2-E(4), 2+E(4) ]
gap> Factors(R,7); Factors(R,11); Factors(R,13);
[ 7 ]
[ 11 ]
[ 3-2*E(4), 3+2*E(4) ]
gap> Factors(R,17); Factors(R,19); Factors(R,23);
[ 4-E(4), 4+E(4) ]
[ 19 ]
[ 23 ]
gap> Factors(R,29); Factors(R,31); Factors(R,37);
[ 5-2*E(4), 5+2*E(4) ]
[ 31 ]
[ 6-E(4), 6+E(4) ]
gap> Factors(R,41); Factors(R,43); Factors(R,47);
[ 5-4*E(4), 5+4*E(4) ]
[ 43 ]
[ 47 ]
gap> Factors(R,53);
[ 7-2*E(4), 7+2*E(4) ]
gap> Factors(R,59);
[ 59 ]

Exercise 16.2
gap> 2 mod 4; 3 mod 4; 5 mod 4;
2
3
1
gap> 7 mod 4; 11 mod 4; 13 mod 4;
3
3
1
gap> 17 mod 4; 19 mod 4; 23 mod 4;
1
3
3

Section 17: Solvable Groups

gap> S:= SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> DerivedSeries(S);
[ Group([ (3,4), (2,4,3), (1,4)(2,3), (1,2)(3,4) ]), Group([ (2,4,3), (1,4)(2,3), (1,2)(3,4) ]), 
  Group([ (1,4)(2,3), (1,2)(3,4) ]), Group(()) ]
gap> d4:= DihedralGroup(IsPermGroup, 8);
Group([ (1,2,3,4), (2,4) ])
gap> DerivedSeries(d4);
[ Group([ (2,4), (1,2,3,4), (1,3)(2,4) ]), Group([ (1,3)(2,4) ]), Group(()) ]
gap> IsSolvable(d4);
true
gap> S:= SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> DerivedSeries(S);
[ Sym( [ 1 .. 5 ] ), Group([ (1,3,2), (2,4,3), (2,3)(4,5) ]) ]
gap> a:= AlternatingGroup(20);
Alt( [ 1 .. 20 ] )
gap> DerivedSeries(a);
[ Alt( [ 1 .. 20 ] ) ]

Exercise 17.2
gap> d5:= DihedralGroup(IsPermGroup, 10);
Group([ (1,2,3,4,5), (2,5)(3,4) ])
gap> DerivedSeries(d5);
[ Group([ (2,5)(3,4), (1,3,5,2,4) ]), Group([ (1,3,5,2,4) ]), Group(()) ]
gap> d30:= DihedralGroup(IsPermGroup, 60);
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), 
  (2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17) ])
gap> DerivedSeries(d30);
[ Group([ (2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17), 
      (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), 
      (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) ]), 
  Group([ (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), 
      (1,7,13,19,25)(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) ]), Group(()) ]

gap> a:= AlternatingGroup(20);
Alt( [ 1 .. 20 ] )
gap> DerivedSeries(a);
[ Alt( [ 1 .. 20 ] ) ]

Exercise 17.4
gap> d4:= DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> d8:= DihedralGroup(IsPermGroup,16);
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
gap> d:=DirectProduct(d4,d8)
> ;
Group([ (1,2,3,4), (2,4), (5,6,7,8,9,10,11,12), (6,12)(7,11)(8,10) ])
gap> Elements(d);
[ (), (6,12)(7,11)(8,10), (5,6)(7,12)(8,11)(9,10), (5,6,7,8,9,10,11,12), (5,7)(8,12)(9,11), 
  (5,7,9,11)(6,8,10,12), (5,8)(6,7)(9,12)(10,11), (5,8,11,6,9,12,7,10), (5,9)(6,8)(10,12), 
  (5,9)(6,10)(7,11)(8,12), (5,10)(6,9)(7,8)(11,12), (5,10,7,12,9,6,11,8), (5,11)(6,10)(7,9), 
  (5,11,9,7)(6,12,10,8), (5,12,11,10,9,8,7,6), (5,12)(6,11)(7,10)(8,9), (2,4), (2,4)(6,12)(7,11)(8,10), 
  (2,4)(5,6)(7,12)(8,11)(9,10), (2,4)(5,6,7,8,9,10,11,12), (2,4)(5,7)(8,12)(9,11), (2,4)(5,7,9,11)(6,8,10,12), 
  (2,4)(5,8)(6,7)(9,12)(10,11), (2,4)(5,8,11,6,9,12,7,10), (2,4)(5,9)(6,8)(10,12), 
  (2,4)(5,9)(6,10)(7,11)(8,12), (2,4)(5,10)(6,9)(7,8)(11,12), (2,4)(5,10,7,12,9,6,11,8), 
  (2,4)(5,11)(6,10)(7,9), (2,4)(5,11,9,7)(6,12,10,8), (2,4)(5,12,11,10,9,8,7,6), (2,4)(5,12)(6,11)(7,10)(8,9), 
  (1,2)(3,4), (1,2)(3,4)(6,12)(7,11)(8,10), (1,2)(3,4)(5,6)(7,12)(8,11)(9,10), (1,2)(3,4)(5,6,7,8,9,10,11,12), 
  (1,2)(3,4)(5,7)(8,12)(9,11), (1,2)(3,4)(5,7,9,11)(6,8,10,12), (1,2)(3,4)(5,8)(6,7)(9,12)(10,11), 
  (1,2)(3,4)(5,8,11,6,9,12,7,10), (1,2)(3,4)(5,9)(6,8)(10,12), (1,2)(3,4)(5,9)(6,10)(7,11)(8,12), 
  (1,2)(3,4)(5,10)(6,9)(7,8)(11,12), (1,2)(3,4)(5,10,7,12,9,6,11,8), (1,2)(3,4)(5,11)(6,10)(7,9), 
  (1,2)(3,4)(5,11,9,7)(6,12,10,8), (1,2)(3,4)(5,12,11,10,9,8,7,6), (1,2)(3,4)(5,12)(6,11)(7,10)(8,9), 
  (1,2,3,4), (1,2,3,4)(6,12)(7,11)(8,10), (1,2,3,4)(5,6)(7,12)(8,11)(9,10), (1,2,3,4)(5,6,7,8,9,10,11,12), 
  (1,2,3,4)(5,7)(8,12)(9,11), (1,2,3,4)(5,7,9,11)(6,8,10,12), (1,2,3,4)(5,8)(6,7)(9,12)(10,11), 
  (1,2,3,4)(5,8,11,6,9,12,7,10), (1,2,3,4)(5,9)(6,8)(10,12), (1,2,3,4)(5,9)(6,10)(7,11)(8,12), 
  (1,2,3,4)(5,10)(6,9)(7,8)(11,12), (1,2,3,4)(5,10,7,12,9,6,11,8), (1,2,3,4)(5,11)(6,10)(7,9), 
  (1,2,3,4)(5,11,9,7)(6,12,10,8), (1,2,3,4)(5,12,11,10,9,8,7,6), (1,2,3,4)(5,12)(6,11)(7,10)(8,9), (1,3), 
  (1,3)(6,12)(7,11)(8,10), (1,3)(5,6)(7,12)(8,11)(9,10), (1,3)(5,6,7,8,9,10,11,12), (1,3)(5,7)(8,12)(9,11), 
  (1,3)(5,7,9,11)(6,8,10,12), (1,3)(5,8)(6,7)(9,12)(10,11), (1,3)(5,8,11,6,9,12,7,10), (1,3)(5,9)(6,8)(10,12), 
  (1,3)(5,9)(6,10)(7,11)(8,12), (1,3)(5,10)(6,9)(7,8)(11,12), (1,3)(5,10,7,12,9,6,11,8), 
  (1,3)(5,11)(6,10)(7,9), (1,3)(5,11,9,7)(6,12,10,8), (1,3)(5,12,11,10,9,8,7,6), (1,3)(5,12)(6,11)(7,10)(8,9), 
  (1,3)(2,4), (1,3)(2,4)(6,12)(7,11)(8,10), (1,3)(2,4)(5,6)(7,12)(8,11)(9,10), (1,3)(2,4)(5,6,7,8,9,10,11,12), 
  (1,3)(2,4)(5,7)(8,12)(9,11), (1,3)(2,4)(5,7,9,11)(6,8,10,12), (1,3)(2,4)(5,8)(6,7)(9,12)(10,11), 
  (1,3)(2,4)(5,8,11,6,9,12,7,10), (1,3)(2,4)(5,9)(6,8)(10,12), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12), 
  (1,3)(2,4)(5,10)(6,9)(7,8)(11,12), (1,3)(2,4)(5,10,7,12,9,6,11,8), (1,3)(2,4)(5,11)(6,10)(7,9), 
  (1,3)(2,4)(5,11,9,7)(6,12,10,8), (1,3)(2,4)(5,12,11,10,9,8,7,6), (1,3)(2,4)(5,12)(6,11)(7,10)(8,9), 
  (1,4,3,2), (1,4,3,2)(6,12)(7,11)(8,10), (1,4,3,2)(5,6)(7,12)(8,11)(9,10), (1,4,3,2)(5,6,7,8,9,10,11,12), 
  (1,4,3,2)(5,7)(8,12)(9,11), (1,4,3,2)(5,7,9,11)(6,8,10,12), (1,4,3,2)(5,8)(6,7)(9,12)(10,11), 
  (1,4,3,2)(5,8,11,6,9,12,7,10), (1,4,3,2)(5,9)(6,8)(10,12), (1,4,3,2)(5,9)(6,10)(7,11)(8,12), 
  (1,4,3,2)(5,10)(6,9)(7,8)(11,12), (1,4,3,2)(5,10,7,12,9,6,11,8), (1,4,3,2)(5,11)(6,10)(7,9), 
  (1,4,3,2)(5,11,9,7)(6,12,10,8), (1,4,3,2)(5,12,11,10,9,8,7,6), (1,4,3,2)(5,12)(6,11)(7,10)(8,9), (1,4)(2,3), 
  (1,4)(2,3)(6,12)(7,11)(8,10), (1,4)(2,3)(5,6)(7,12)(8,11)(9,10), (1,4)(2,3)(5,6,7,8,9,10,11,12), 
  (1,4)(2,3)(5,7)(8,12)(9,11), (1,4)(2,3)(5,7,9,11)(6,8,10,12), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11), 
  (1,4)(2,3)(5,8,11,6,9,12,7,10), (1,4)(2,3)(5,9)(6,8)(10,12), (1,4)(2,3)(5,9)(6,10)(7,11)(8,12), 
  (1,4)(2,3)(5,10)(6,9)(7,8)(11,12), (1,4)(2,3)(5,10,7,12,9,6,11,8), (1,4)(2,3)(5,11)(6,10)(7,9), 
  (1,4)(2,3)(5,11,9,7)(6,12,10,8), (1,4)(2,3)(5,12,11,10,9,8,7,6), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9) ]
gap> DerivedSeries(d);
[ Group([ (6,12)(7,11)(8,10), (1,2,3,4), (2,4), (5,6,7,8,9,10,11,12), (5,7,9,11)(6,8,10,12), (1,3)(2,4) ]), 
  Group([ (5,7,9,11)(6,8,10,12), (5,9)(6,10)(7,11)(8,12), (1,3)(2,4) ]), Group(()) ]
gap> IsSolvable(d);
true

Section 18: Orbits and Stabilizers

gap> G := SymmetricGroup(8);
Sym( [ 1 .. 8 ] )
gap> a := (1,2,3)(4,5,6);
(1,2,3)(4,5,6)
gap> b := (7,8);
(7,8)
gap> H := Subgroup(G,[a,b]);
Group([ (1,2,3)(4,5,6), (7,8) ])
gap> Elements(H);
[ (), (7,8), (1,2,3)(4,5,6), (1,2,3)(4,5,6)(7,8), (1,3,2)(4,6,5), (1,3,2)(4,6,5)(7,8) ]
gap> Orbit(H,1);
[ 1, 3, 2 ]
gap> Orbit(H,7);
[ 7, 8 ]
gap> Stabilizer(H,1);
Group([ (7,8) ])
gap> Order(Stabilizer(H,1));
2
gap> Stabilizer(H,7);
Group([ (1,2,3)(4,5,6) ])
gap> Order(Stabilizer(H,7));
3
gap> Elements(Stabilizer(H,7));
[ (), (1,2,3)(4,5,6), (1,3,2)(4,6,5) ]

Exercise 18.1
gap> G := DihedralGroup(IsPermGroup,20);
Group([ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7) ])
gap> Size(Orbit(G,1));
10
gap> Size(Orbit(G,2));
10
gap> Size(Orbit(G,3));
10

Exercise 18.2
gap> Order(Stabilizer(G,1));
2
gap> Order(Stabilizer(G,2));
2
gap> Order(Stabilizer(G,3));
2

Exercise 18.3
gap> G := DihedralGroup(IsPermGroup,98);
<permutation group with 2 generators>
gap> Size(Orbit(G,1));
49
gap> Order(Stabilizer(G,1));
2
gap> G := DihedralGroup(IsPermGroup,100);
<permutation group with 2 generators>
gap> Size(Orbit(G,1));
50
gap> Order(Stabilizer(G,1));
2

Exercise 18.6
gap> G := SymmetricGroup(12);
Sym( [ 1 .. 12 ] )
gap> H := Subgroup(G,[(1,2,3),(4,5)(6,7),(8,9)(9,10),(9,10)(11,12)]);
Permutation: cycles must be disjoint and duplicate-free
gap> G := SymmetricGroup(13);
Sym( [ 1 .. 13 ] )
gap> H := Subgroup(G,[(1,2,3),(4,5)(6,7),(8,9)(10,11),(10,11)(12,13)]);
Group([ (1,2,3), (4,5)(6,7), (8,9)(10,11), (10,11)(12,13) ])
gap> Size(Orbit(H,1));
3
gap> Order(Stabilizer(H,1));
8
gap> Size(Orbit(H,4));
2
gap> Order(Stabilizer(H,4));
12
gap> Size(Orbit(H,10));
2
gap> Order(Stabilizer(H,10));
12
gap> Order(H);
24
gap> quit;

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