Section 14, continued
gap> DegreeFFE(Z(2^4)^1); DegreeFFE(Z(2^4)^2); DegreeFFE(Z(2^4)^3); 4 4 4 gap> DegreeFFE(Z(2^4)^4); DegreeFFE(Z(2^4)^5); DegreeFFE(Z(2^4)^6); 4 2 4 gap> DegreeFFE(Z(2^4)^7); DegreeFFE(Z(2^4)^8); DegreeFFE(Z(2^4)^9); 4 4 4 gap> DegreeFFE(Z(2^4)^10); 2 gap> Order(Z(2^4)^1); Order(Z(2^4)^2); Order(Z(2^4)^3); 15 15 5 gap> Order(Z(2^4)^4); Order(Z(2^4)^5); Order(Z(2^4)^6); 15 3 5 gap> Order(Z(2^4)^7); Order(Z(2^4)^8); Order(Z(2^4)^9); 15 15 5 gap> Order(Z(2^4)^10); 3 gap> x:= X(GF(3),"x"); x gap> Factors(x^(3^2)-x); [ x, x+Z(3)^0, x-Z(3)^0, x^2+Z(3)^0, x^2+x-Z(3)^0, x^2-x-Z(3)^0 ] gap> Factors(x^(3^3)-x); [ x, x+Z(3)^0, x-Z(3)^0, x^3-x+Z(3)^0, x^3-x-Z(3)^0, x^3+x^2-Z(3)^0, x^3+x^2+x-Z(3)^0, x^3+x^2-x+Z(3)^0, x^3-x^2+Z(3)^0, x^3-x^2+x+Z(3)^0, x^3-x^2-x-Z(3)^0 ] gap> Factors(x^(3^4)-x); [ x, x+Z(3)^0, x-Z(3)^0, x^2+Z(3)^0, x^2+x-Z(3)^0, x^2-x-Z(3)^0, x^4+x-Z(3)^0, x^4-x-Z(3)^0, x^4+x^2-Z(3)^0, x^4+x^2+x+Z(3)^0, x^4+x^2-x+Z(3)^0, x^4-x^2-Z(3)^0, x^4+x^3-Z(3)^0, x^4+x^3-x+Z(3)^0, x^4+x^3+x^2+Z(3)^0, x^4+x^3+x^2+x+Z(3)^0, x^4+x^3+x^2-x-Z(3)^0, x^4+x^3-x^2-x-Z(3)^0, x^4-x^3-Z(3)^0, x^4-x^3+x+Z(3)^0, x^4-x^3+x^2+Z(3)^0, x^4-x^3+x^2+x-Z(3)^0, x^4-x^3+x^2-x+Z(3)^0, x^4-x^3-x^2+x-Z(3)^0 ] gap> x:= X(GF(5),"x"); x gap> Factors(x^(5^2)-x); [ x, x+Z(5)^0, x+Z(5), x-Z(5)^0, x+Z(5)^3, x^2+Z(5), x^2+Z(5)^3, x^2+x+Z(5)^0, x^2+x+Z(5), x^2+Z(5)*x-Z(5)^0, x^2+Z(5)*x+Z(5)^3, x^2-x+Z(5)^0, x^2-x+Z(5), x^2+Z(5)^3*x-Z(5)^0, x^2+Z(5)^3*x+Z(5)^3 ] gap> Factors(x^(5^3)-x); [ x, x+Z(5)^0, x+Z(5), x-Z(5)^0, x+Z(5)^3, x^3+x+Z(5)^0, x^3+x-Z(5)^0, x^3+Z(5)*x+Z(5)^0, x^3+Z(5)*x-Z(5)^0, x^3-x+Z(5), x^3-x+Z(5)^3, x^3+Z(5)^3*x+Z(5), x^3+Z(5)^3*x+Z(5)^3, x^3+x^2+Z(5)^0, x^3+x^2+Z(5), x^3+x^2+x-Z(5)^0, x^3+x^2+x+Z(5)^3, x^3+x^2-x+Z(5)^0, x^3+x^2-x+Z(5)^3, x^3+x^2+Z(5)^3*x+Z(5)^0, x^3+x^2+Z(5)^3*x-Z(5)^0, x^3+Z(5)*x^2+Z(5)^0, x^3+Z(5)*x^2+Z(5)^3, x^3+Z(5)*x^2+x-Z(5)^0, x^3+Z(5)*x^2+x+Z(5)^3, x^3+Z(5)*x^2+Z(5)*x+Z(5), x^3+Z(5)*x^2+Z(5)*x+Z(5)^3, x^3+Z(5)*x^2-x+Z(5), x^3+Z(5)*x^2-x-Z(5)^0, x^3-x^2-Z(5)^0, x^3-x^2+Z(5)^3, x^3-x^2+x+Z(5)^0, x^3-x^2+x+Z(5), x^3-x^2-x+Z(5), x^3-x^2-x-Z(5)^0, x^3-x^2+Z(5)^3*x+Z(5)^0, x^3-x^2+Z(5)^3*x-Z(5)^0, x^3+Z(5)^3*x^2+Z(5), x^3+Z(5)^3*x^2-Z(5)^0, x^3+Z(5)^3*x^2+x+Z(5)^0, x^3+Z(5)^3*x^2+x+Z(5), x^3+Z(5)^3*x^2+Z(5)*x+Z(5), x^3+Z(5)^3*x^2+Z(5)*x+Z(5)^3, x^3+Z(5)^3*x^2-x+Z(5)^0, x^3+Z(5)^3*x^2-x+Z(5)^3 ] gap> x:= X(GF(2),"x"); x gap> IsIrreducible(x^5+x^4+x+1); false gap> IsIrreducible(x^5+x+1); false gap> IsIrreducible(x^5+x^2+1); true gap> F:= AlgebraicExtension(GF(2), x^5+x^2+1);
Project by Bill and Edward
gap> g:= GL(3, GF(2)); SL(3,2) gap> g:= GL(3, GF(2)); SL(3,2) gap> Order(g); 168 gap> SylowSubgroup(g, 2);gap> h:=SylowSubgroup(g, 2); gap> ConjugateSubgroups(g,h); [ , , , , , , , , , , , , , , , , , , , , ] gap> Size(ConjugateSubgroups(g,h)); 21 gap> Read("AMSylow.txt"); gap> Sylow(168,2); [ 1, 3, 7, 21 ] gap> Factors(168); [ 2, 2, 2, 3, 7 ] gap> Sylow(168,3); [ 1, 4, 7, 28 ] gap> Sylow(168,7); [ 1, 8 ] gap> Factors(84); [ 2, 2, 3, 7 ] gap> Sylow(84,2); Sylow(84,3); Sylow(84,7); [ 1, 3, 7, 21 ] [ 1, 4, 7, 28 ] [ 1 ] gap> m:= MagmaByMultiplication([[1,2], [2,1]]); Variable: 'MagmaByMultiplication' must have a value gap> m:= MagmaByMultiplicationTable([[1,2], [2,1]]); gap> Read("Assoc"); Error, file "Assoc" must exist and be readable called from ( ) called from read-eval-loop Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk> quit; gap> Read("Assoc.txt"); gap> Print("Assoc.txt"); Assoc.txtgap> Print(Assoc); function ( m ) local x, y, z; for x in Elements( m ) do for y in Elements( m ) do for z in Elements( m ) do Print( x, "* (", y, "*", z, ") = (", x, "*", y, ") *", z, "\n" ); Print( x * (y * z) = x * y * z, "\n" ); od; od; od; return; endgap> Assoc(g); [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]* ( [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]* [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]) = ( [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]* [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]) * [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ] true ... ( [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]* [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, Z(2)^0, 0*Z(2) ] ]) * [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ] ] true [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]* ( user interrupt at Print( "\<,\< \>\>" ); called from Print( x, "* (", y, "*", z, ") = (", x, "*", y, ") *", z, "\n" ); called from ( ) called from read-eval-loop Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk> quit; [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2)gap> Assoc(m); m1* (m1*m1) = (m1*m1) *m1 true m1* (m1*m2) = (m1*m1) *m2 true m1* (m2*m1) = (m1*m2) *m1 true m1* (m2*m2) = (m1*m2) *m2 true m2* (m1*m1) = (m2*m1) *m1 true m2* (m1*m2) = (m2*m1) *m2 true m2* (m2*m1) = (m2*m2) *m1 true m2* (m2*m2) = (m2*m2) *m2 true gap> Read("Ident.txt"); gap> Print(Ident); function ( n ) local x, y, l; l := [ ]; for x in Elements( n ) do for y in Elements( n ) do if x * y = y then Add( l, y ); fi; Print( x, "*", y, "=", x * y, "\n" ); od; if Elements( l ) = Elements( n ) then Print( x, " is an identity", "\n" ); else Print( x, " is not an identity", "\n" ); fi; l := [ ]; od; return; endgap> Ident(m); m1*m1=m1 m1*m2=m2 m1 is an identity m2*m1=m2 m2*m2=m1 m2 is not an identity gap> Read("Inver.txt"); gap> Print(Inver); function ( m, e ) local x, y, k; k := MagmaElement( m, e ); for x in Elements( m ) do for y in Elements( m ) do if x * y = k then Print( "The inverse to ", x, " is ", y, " since ", x, " * ", y, " = ", k, "\n" ); fi; od; od; return; endgap> Inver(m,1); The inverse to m1 is m1 since m1 * m1 = m1 The inverse to m2 is m2 since m2 * m2 = m1 gap> IsGroup(m); false gap> AsGroup(m); gap> j:=AsGroup(m); gap> IsGroup(j); true gap> Read("Movement.txt"); gap> Movement((1,2,3), 2); (1,2,3) sends 1 to 2 (1,2,3) sends 2 to 3 gap> Movement((1,3)(4,6,8), 3); (1,3)(4,6,8) sends 1 to 3 (1,3)(4,6,8) sends 2 to 2 (1,3)(4,6,8) sends 3 to 1 gap> Movement((1,3)(4,6,8), 9); (1,3)(4,6,8) sends 1 to 3 (1,3)(4,6,8) sends 2 to 2 (1,3)(4,6,8) sends 3 to 1 (1,3)(4,6,8) sends 4 to 6 (1,3)(4,6,8) sends 5 to 5 (1,3)(4,6,8) sends 6 to 8 (1,3)(4,6,8) sends 7 to 7 (1,3)(4,6,8) sends 8 to 4 (1,3)(4,6,8) sends 9 to 9 gap> Movement((1,3,4), 5); (1,3,4) sends 1 to 3 (1,3,4) sends 2 to 2 (1,3,4) sends 3 to 4 (1,3,4) sends 4 to 1 (1,3,4) sends 5 to 5 gap> Movement((1,2,4), 5); (1,2,4) sends 1 to 2 (1,2,4) sends 2 to 4 (1,2,4) sends 3 to 3 (1,2,4) sends 4 to 1 (1,2,4) sends 5 to 5 gap> Movement((3,2,4), 5); (2,4,3) sends 1 to 1 (2,4,3) sends 2 to 4 (2,4,3) sends 3 to 2 (2,4,3) sends 4 to 3 (2,4,3) sends 5 to 5 gap> Read("CountMovement"); Error, file "CountMovement" must exist and be readable called from ( ) called from read-eval-loop Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk> quit; gap> Read("CountMovement.txt"); gap> Print(CountMovement); function ( s, n ) local x, y, x1, y1, i; i := 0; for x in Elements( [ 1 .. n ] ) do for y in Elements( [ x + 1 .. n ] ) do x1 := x ^ s; y1 := y ^ s; Print( x, y, x ^ s, y ^ s, "\n" ); if y1 < x1 then i := i + 1; fi; od; od; Print( s, " switches ", i, " pairs ", "\n" ); return; endgap> CountMovement((1,3,4),5); 1232 1334 1431 1535 2324 2421 2525 3441 3545 4515 (1,3,4) switches 4 pairs gap> CountMovement((1,2,4),5); 1224 1323 1421 1525 2343 2441 2545 3431 3535 4515 (1,2,4) switches 4 pairs gap> CountMovement((1,2,3),5); 1223 1321 1424 1525 2331 2434 2535 3414 3515 4545 (1,2,3) switches 2 pairs gap> CountMovement((4,2,3),5); 1213 1314 1412 1515 2334 2432 2535 3442 3545 4525 (2,3,4) switches 2 pairs gap> CountMovement; function( s, n ) ... end
Project by Chiru and Jen
gap> List(AllSmallGroups(16), StructureDescription);
[ "C16", "C4 x C4", "(C4 x C2) : C2", "C4 : C4", "C8 x C2", "C8 : C2", "D16", "QD16", "Q16", "C4 x C2 x C2", "C2 x D8",
"C2 x Q8", "(C4 x C2) : C2", "C2 x C2 x C2 x C2" ]
gap> G:=AllSmallGroups(16)[1];
gap> elts:=Elements(G);
[ of ..., f1, f2, f3, f4, f1*f2, f1*f3, f1*f4, f2*f3, f2*f4, f3*f4, f1*f2*f3, f1*f2*f4, f1*f3*f4, f2*f3*f4,
f1*f2*f3*f4 ]
gap> G;
gap> MinimalGeneratingSet(G);
[ f1 ]
gap> H:=Subgroup(G,[elts[2]]);
Group([ f1 ])
gap> H=G;
true
gap> c16:=[];
[ ]
gap> for i in [1..16] do
> x:=Factorization(H,elts[i]);
> Append(c16,[x]);
> od;
gap> c16;
[ , x1, x1^2, x1^4, x1^8, x1^3, x1^5, x1^-7, x1^6, x1^-6, x1^-4, x1^7, x1^-5, x1^-3, x1^-2, x1^-1 ]
gap> list1:=[1,2,3,4,5,6];
[ 1, 2, 3, 4, 5, 6 ]
gap> D:=[];DInv:=[];
[ ]
[ ]
gap> for i in [1..6] do
> x:=c16[list1[i]];
> Append(D,[x]);
> do;
Syntax error: od expected
do;
^
gap> for i in [1..6] do
> x:=c16[list1[i]];
> Append(D,[x]);
> od;
gap> D;
[ , x1, x1^2, x1^4, x1^8, x1^3 ]
gap> for i in [1..6] do
> y:=Inverse(D[i]);
> Append(DInv,[y]);
> od;
gap> DInv;
[ , x1^-1, x1^-2, x1^-4, x1^-8, x1^-3 ]
gap> S:=[];
[ ]
gap> for i in [1..6] do
> for j in [1..6] do
> z:=D[i]*DInv[j];
> Append(S,[z]);
> od;
> od;
gap> Size(S);
36
gap> Elements(S);
[ , x1^-1, x1, x1^-2, x1^2, x1^-3, x1^3, x1^-4, x1^4, x1^-5, x1^5, x1^-6, x1^6, x1^-7, x1^7, x1^-8, x1^8 ]
gap> S;
[ , x1^-1, x1^-2, x1^-4, x1^-8, x1^-3, x1, , x1^-1, x1^-3, x1^-7, x1^-2, x1^2, x1,
, x1^-2, x1^-6, x1^-1, x1^4, x1^3, x1^2, , x1^-4, x1, x1^8, x1^7, x1^6, x1^4,
, x1^5, x1^3, x1^2, x1, x1^-1, x1^-5, ]
gap>
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This PREP workshop is made possible by the NSF grant DUE: 0341481