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What C# can do for studying Finite Groups, quotient groups, semi-direct products, homomorphisms, automorphisms group, characters table, minimalistic rings and fields manipulations, polynomials factoring, fields extensions and many more...
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What C# can do for studying Finite Groups, abelians or not, quotient groups, direct products or semi-direct products, homomorphisms, automorphisms group, with minimalistic manipulations for rings, fields, numbers, polynomials factoring, field extensions, algebraics numbers and many more...
Searching semi-direct product
GlobalStopWatch.Restart();vars7=newSn(7);vara=s7[(1,2,3,4,5,6,7)];varallC3=s7.Where(p=>(p^3)==s7.Neutral()).ToArray();varb=allC3.First(p=>Group.GenerateElements(s7,a,p).Count()==21);GlobalStopWatch.Stop();Console.WriteLine("|S7|={0}, |{{b in S7 with b^3 = 1}}| = {1}",s7.Count(),allC3.Count());Console.WriteLine("First Solution |HK| = 21 : h = {0} and k = {1}",a,b);Console.WriteLine();varh=Group.Generate("H",s7,a);varg21=Group.Generate("G21",s7,a,b);DisplayGroup.Head(g21);DisplayGroup.Head(g21.Over(h));GlobalStopWatch.Show("Group21");
will output
|S7|=5040, |{b in S7 with b^3 = 1}| = 351First Solution |HK| = 21 : h = [(1 2 3 4 5 6 7)] and k = [(2 3 5)(4 7 6)]|G21| = 21Type NonAbelianGroupBaseGroup S7|G21/H| = 3Type AbelianGroupBaseGroup G21/HGroup |G21| = 21NormalSubGroup |H| = 7# Group21 Time:113 msComparing the previous results with the group presented by
GlobalStopWatch.Restart();varwg=newWordGroup("a7, b3, a2 = bab-1");GlobalStopWatch.Stop();DisplayGroup.Head(wg);varn=Group.Generate("<a>",wg,wg["a"]);DisplayGroup.Head(wg.Over(n));GlobalStopWatch.Show($"{wg}");Console.WriteLine();
will produce
|WG[a,b]| = 21Type NonAbelianGroupBaseGroup WG[a,b]|WG[a,b]/<a>| = 3Type AbelianGroupBaseGroup WG[a,b]/<a>Group |WG[a,b]| = 21NormalSubGroup |<a>| = 7# WG[a,b] Time:42 msAnother way for the previous example
GlobalStopWatch.Restart();varc7=newCn(7);varc3=newCn(3);varg21=Group.SemiDirectProd(c7,c3);GlobalStopWatch.Stop();varn=Group.Generate("N",g21,g21[1,0]);DisplayGroup.HeadSdp(g21);DisplayGroup.Head(g21.Over(n));GlobalStopWatch.Show("Group21");
will output
|C7 x: C3| = 21Type NonAbelianGroupBaseGroup C7 x C3NormalGroup |C7| = 7ActionGroup |C3| = 3Action FaithFullg=0 y(g) = (0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6)g=1 y(g) = (0->0, 1->2, 2->4, 3->6, 4->1, 5->3, 6->5)g=2 y(g) = (0->0, 1->4, 2->1, 3->5, 4->2, 5->6, 6->3)|(C7 x: C3)/N| = 3Type AbelianGroupBaseGroup (C7 x: C3)/NGroup |C7 x: C3| = 21NormalSubGroup |N| = 7# Group21 Time:1 msExpressing the group
vargl27=FG.GL2p(7);varg21mat=Group.IsomorphicSubgroup(gl27,g21);DisplayGroup.HeadGenerators(g21mat);
will output
|C7 x: C3| = 21Type NonAbelianGroupBaseGroup GL(2,7)SuperGroup |GL(2,7)| = 2016Generators of C7 x: C3gen1 of order 3[2, 0][0, 1]gen2 of order 7[1, 1][0, 1]Displaying characters table for the group
FG.CharacterTable(g21).DisplayCells();
will output
|C7 x: C3| = 21Type NonAbelianGroupBaseGroup C7 x C3[Class 1 3a 3b 7a 7b][ Size 1 7 7 3 3][ ][ Ꭓ.1 1 1 1 1 1][ Ꭓ.2 1 ξ3 ξ3² 1 1][ Ꭓ.3 1 ξ3² ξ3 1 1][ Ꭓ.4 3 0 0 -1/2 - 1/2·I√7 -1/2 + 1/2·I√7][ Ꭓ.5 3 0 0 -1/2 + 1/2·I√7 -1/2 - 1/2·I√7]All i, Sum[g](Xi(g)Xi(g^−1)) = |G| : TrueAll i <> j, Sum[g](Xi(g)Xj(g^−1)) = 0 : TrueAll g, h in Cl(g), Sum[r](Xr(g)Xr(h^−1)) = |Cl(g)| : TrueAll g, h not in Cl(g), Sum[r](Xr(g)Xr(h^−1)) = 0 : TrueComputing Galois Group of irreductible polynomial
Ring.DisplayPolynomial=MonomDisplay.StarCaret;varx=FG.QPoly('X');varP=x.Pow(7)-8*x.Pow(5)-2*x.Pow(4)+16*x.Pow(3)+6*x.Pow(2)-6*x-2;// GroupNames websiteGaloisApplicationsPart2.GaloisGroupChebotarev(P,details:true);
will output
f = X^7 - 8*X^5 - 2*X^4 + 16*X^3 + 6*X^2 - 6*X - 2Disc(f) = 1817487424 ~ 2^6 * 73^4#1 P = 3 shape (7)#2 P = 5 shape (1, 3, 3)#3 P = 7 shape (7)#4 P = 11 shape (1, 3, 3)#5 P = 13 shape (1, 3, 3)#6 P = 17 shape (7)#7 P = 19 shape (1, 3, 3)#8 P = 23 shape (1, 3, 3)#9 P = 29 shape (1, 3, 3)#10 P = 31 shape (1, 3, 3)actual types [(7), 3] [(1, 3, 3), 7]expected types [(7), 6] [(1, 3, 3), 14] [(1, 1, 1, 1, 1, 1, 1), 1]Distances { Name = F_21(7) = 7:3, order = 21, dist = 0.6666666666666664 } { Name = F_42(7) = 7:6, order = 42, dist = 8.8 } { Name = L(7) = L(3,2), order = 168, dist = 25.6 } { Name = A7, order = 2520, dist = 380 } { Name = S7, order = 5040, dist = 538.6666666666666 }P = X^7 - 8*X^5 - 2*X^4 + 16*X^3 + 6*X^2 - 6*X - 2Gal(P) = F_21(7) = 7:3|F_21(7) = 7:3| = 21Type NonAbelianGroupBaseGroup S7SuperGroup |Symm7| = 5040DisplayGroup.HeadElementsCayleyGraph(g21,gens:[g21["a"],g21["b-1"]]);
will output
Galois Group of polynomial
Ring.DisplayPolynomial=MonomDisplay.StarCaret;varx=FG.QPoly('X');varP=x.Pow(5)+x.Pow(4)-4*x.Pow(3)-3*x.Pow(2)+3*x+1;varroots=IntFactorisation.AlgebraicRoots(P,details:true);vargal=GaloisTheory.GaloisGroup(roots,details:true);DisplayGroup.AreIsomorphics(gal,FG.Abelian(5));
will output
[...]f = X^5 + X^4 - 4*X^3 - 3*X^2 + 3*X + 1 with f(y) = 0Square free norm : Norm(f(X + 2*y)) = x^25 - 5*x^24 - 98*x^23 + 503*x^22 + 3916*x^21 - 20988*x^20 - 82808*x^19 + 476245*x^18 + 1001759*x^17 - 6482223*x^16 - 6888926*x^15 + 55077950*x^14 + 23535811*x^13 - 295014199*x^12 - 8852570*x^11 + 984611573*x^10 - 207998384*x^9 - 1981105500*x^8 + 676565912*x^7 + 2253157335*x^6 - 871099834*x^5 - 1278826318*x^4 + 467945878*x^3 + 268636799*x^2 - 89574789*x + 1746623 = (x^5 - x^4 - 26*x^3 + 47*x^2 + 47*x - 1) * (x^5 - x^4 - 26*x^3 + 25*x^2 + 91*x - 67) * (x^5 - x^4 - 26*x^3 + 25*x^2 + 157*x - 199) * (x^5 - x^4 - 26*x^3 - 19*x^2 + 113*x + 131) * (x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1)X^5 + X^4 - 4*X^3 - 3*X^2 + 3*X + 1 = (X + y^4 + y^3 - 3*y^2 - 2*y + 1) * (X - y^4 + 4*y^2 - 2) * (X - y^3 + 3*y) * (X - y^2 + 2) * (X - y)Are equals TruePolynomial P = X^5 + X^4 - 4*X^3 - 3*X^2 + 3*X + 1Factorization in Q(α)[X] with P(α) = 0 X - α X - α^2 + 2 X - α^3 + 3*α X + α^4 + α^3 - 3*α^2 - 2*α + 1 X - α^4 + 4*α^2 - 2|Gal( Q(α)/Q )| = 5Type AbelianGroupBaseGroup S5Elements(1)[1] = [](2)[5] = [(1 2 5 3 4)](3)[5] = [(1 3 2 4 5)](4)[5] = [(1 4 3 5 2)](5)[5] = [(1 5 4 2 3)]Gal( Q(α)/Q ) IsIsomorphicTo C5 : TrueComputing
Ring.DisplayPolynomial=MonomDisplay.Caret;varx=FG.QPoly('X');var(X, _)=FG.EPolyXc(x.Pow(2)-2,'a');var(minPoly,a0,b0)=IntFactorisation.PrimitiveElt(X.Pow(2)-3);varroots=IntFactorisation.AlgebraicRoots(minPoly);Console.WriteLine("Q(√2, √3) = Q(α)");vargal=GaloisTheory.GaloisGroup(roots,details:true);DisplayGroup.AreIsomorphics(gal,FG.Abelian(2,2));
will output
Q(√2, √3) = Q(α)Polynomial P = X^4 - 10*X^2 + 1Factorization in Q(α)[X] with P(α) = 0 X + α X - α X + α^3 - 10*α X - α^3 + 10*α|Gal( Q(α)/Q )| = 4Type AbelianGroupBaseGroup S4Elements(1)[1] = [](2)[2] = [(1 2)(3 4)](3)[2] = [(1 3)(2 4)](4)[2] = [(1 4)(2 3)]Gal( Q(α)/Q ) IsIsomorphicTo C2 x C2 : TrueWith
Ring.DisplayPolynomial=MonomDisplay.Caret;varx=FG.QPoly('X');var(X,i)=FG.EPolyXc(x.Pow(2)+1,'i');var(minPoly, _, _)=IntFactorisation.PrimitiveElt(X.Pow(4)-2);varroots=IntFactorisation.AlgebraicRoots(minPoly);Console.WriteLine("With α^4-2 = 0, Q(α, i) = Q(β)");vargal=GaloisTheory.GaloisGroup(roots,primEltChar:'β',details:true);DisplayGroup.AreIsomorphics(gal,FG.Dihedral(4));
will output
With α^4-2 = 0, Q(α, i) = Q(β)Polynomial P = X^8 + 4*X^6 + 2*X^4 + 28*X^2 + 1Factorization in Q(β)[X] with P(β) = 0 X + β X - β X + 5/12*β^7 + 19/12*β^5 + 5/12*β^3 + 139/12*β X + 5/24*β^7 - 1/24*β^6 + 19/24*β^5 - 5/24*β^4 + 5/24*β^3 - 13/24*β^2 + 127/24*β - 29/24 X + 5/24*β^7 + 1/24*β^6 + 19/24*β^5 + 5/24*β^4 + 5/24*β^3 + 13/24*β^2 + 127/24*β + 29/24 X - 5/24*β^7 - 1/24*β^6 - 19/24*β^5 - 5/24*β^4 - 5/24*β^3 - 13/24*β^2 - 127/24*β - 29/24 X - 5/24*β^7 + 1/24*β^6 - 19/24*β^5 + 5/24*β^4 - 5/24*β^3 + 13/24*β^2 - 127/24*β + 29/24 X - 5/12*β^7 - 19/12*β^5 - 5/12*β^3 - 139/12*β|Gal( Q(β)/Q )| = 8Type NonAbelianGroupBaseGroup S8Elements(1)[1] = [](2)[2] = [(1 2)(3 8)(4 7)(5 6)](3)[2] = [(1 3)(2 8)(4 6)(5 7)](4)[2] = [(1 4)(2 6)(3 7)(5 8)](5)[2] = [(1 5)(2 7)(3 6)(4 8)](6)[2] = [(1 8)(2 3)(4 5)(6 7)](7)[4] = [(1 6 8 7)(2 4 3 5)](8)[4] = [(1 7 8 6)(2 5 3 4)]Gal( Q(β)/Q ) IsIsomorphicTo D8 : True
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What C# can do for studying Finite Groups, quotient groups, semi-direct products, homomorphisms, automorphisms group, characters table, minimalistic rings and fields manipulations, polynomials factoring, fields extensions and many more...
