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();
var s7 = new Sn(7);
var a = s7[(1, 2, 3, 4, 5, 6, 7)];
var allC3 = s7.Where(p => (p ^ 3) == s7.Neutral()).ToArray();
var b = 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();
var h = Group.Generate("H", s7, a);
var g21 = 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}| = 351
First Solution |HK| = 21 : h = [(1 2 3 4 5 6 7)] and k = [(2 3 5)(4 7 6)]
|G21| = 21
Type NonAbelianGroup
BaseGroup S7
|G21/H| = 3
Type AbelianGroup
BaseGroup G21/H
Group |G21| = 21
NormalSubGroup |H| = 7
# Group21 Time:113 ms
Comparing the previous results with the group presented by
GlobalStopWatch.Restart();
var wg = new WordGroup("a7, b3, a2 = bab-1");
GlobalStopWatch.Stop();
DisplayGroup.Head(wg);
var n = Group.Generate("<a>", wg, wg["a"]);
DisplayGroup.Head(wg.Over(n));
GlobalStopWatch.Show($"{wg}");
Console.WriteLine();will produce
|WG[a,b]| = 21
Type NonAbelianGroup
BaseGroup WG[a,b]
|WG[a,b]/<a>| = 3
Type AbelianGroup
BaseGroup WG[a,b]/<a>
Group |WG[a,b]| = 21
NormalSubGroup |<a>| = 7
# WG[a,b] Time:42 ms
Another way for the previous example
GlobalStopWatch.Restart();
var c7 = new Cn(7);
var c3 = new Cn(3);
var g21 = Group.SemiDirectProd(c7, c3);
GlobalStopWatch.Stop();
var n = Group.Generate("N", g21, g21[1, 0]);
DisplayGroup.HeadSdp(g21);
DisplayGroup.Head(g21.Over(n));
GlobalStopWatch.Show("Group21");will output
|C7 x: C3| = 21
Type NonAbelianGroup
BaseGroup C7 x C3
NormalGroup |C7| = 7
ActionGroup |C3| = 3
Action FaithFull
g=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| = 3
Type AbelianGroup
BaseGroup (C7 x: C3)/N
Group |C7 x: C3| = 21
NormalSubGroup |N| = 7
# Group21 Time:1 ms
Expressing the group
var gl27 = FG.GL2p(7);
var g21mat = Group.IsomorphicSubgroup(gl27, g21);
DisplayGroup.HeadGenerators(g21mat);will output
|C7 x: C3| = 21
Type NonAbelianGroup
BaseGroup GL(2,7)
SuperGroup |GL(2,7)| = 2016
Generators of C7 x: C3
gen1 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| = 21
Type NonAbelianGroup
BaseGroup 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| : True
All i <> j, Sum[g](Xi(g)Xj(g^−1)) = 0 : True
All g, h in Cl(g), Sum[r](Xr(g)Xr(h^−1)) = |Cl(g)| : True
All g, h not in Cl(g), Sum[r](Xr(g)Xr(h^−1)) = 0 : True
Computing Galois Group of irreductible polynomial
Ring.DisplayPolynomial = MonomDisplay.StarCaret;
var x = FG.QPoly('X');
var P = x.Pow(7) - 8 * x.Pow(5) - 2 * x.Pow(4) + 16 * x.Pow(3) + 6 * x.Pow(2) - 6 * x - 2; // GroupNames website
GaloisApplicationsPart2.GaloisGroupChebotarev(P, details: true);will output
f = X^7 - 8*X^5 - 2*X^4 + 16*X^3 + 6*X^2 - 6*X - 2
Disc(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 - 2
Gal(P) = F_21(7) = 7:3
|F_21(7) = 7:3| = 21
Type NonAbelianGroup
BaseGroup S7
SuperGroup |Symm7| = 5040
DisplayGroup.HeadElementsCayleyGraph(g21, gens: [g21["a"], g21["b-1"]]);will output
Galois Group of polynomial
Ring.DisplayPolynomial = MonomDisplay.StarCaret;
var x = FG.QPoly('X');
var P = x.Pow(5) + x.Pow(4) - 4 * x.Pow(3) - 3 * x.Pow(2) + 3 * x + 1;
var roots = IntFactorisation.AlgebraicRoots(P, details: true);
var gal = 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) = 0
Square 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 True
Polynomial P = X^5 + X^4 - 4*X^3 - 3*X^2 + 3*X + 1
Factorization 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 )| = 5
Type AbelianGroup
BaseGroup S5
Elements
(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 : True
Computing
Ring.DisplayPolynomial = MonomDisplay.Caret;
var x = FG.QPoly('X');
var (X, _) = FG.EPolyXc(x.Pow(2) - 2, 'a');
var (minPoly, a0, b0) = IntFactorisation.PrimitiveElt(X.Pow(2) - 3);
var roots = IntFactorisation.AlgebraicRoots(minPoly);
Console.WriteLine("Q(√2, √3) = Q(α)");
var gal = 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 + 1
Factorization 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 )| = 4
Type AbelianGroup
BaseGroup S4
Elements
(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 : True
With
Ring.DisplayPolynomial = MonomDisplay.Caret;
var x = FG.QPoly('X');
var (X, i) = FG.EPolyXc(x.Pow(2) + 1, 'i');
var (minPoly, _, _) = IntFactorisation.PrimitiveElt(X.Pow(4) - 2);
var roots = IntFactorisation.AlgebraicRoots(minPoly);
Console.WriteLine("With α^4-2 = 0, Q(α, i) = Q(β)");
var gal = 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 + 1
Factorization 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 )| = 8
Type NonAbelianGroup
BaseGroup S8
Elements
(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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