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aidevnn / FastGoat

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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finite-groupsabelian-groupsquotient-groupspermutationssemi-direct-productcsharphomomorphismsautomorphism-groupgalois-grouppolynomial-factoringfinite-fieldstodd-coxeterfield-extensionsgrobner-basischaracter-tablefield-theorygroup-theorymultiprecisionbch-codescohomology

FastGoat

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...

Example

Searching semi-direct product $C_7 \rtimes C_3$ in $\textbf{S}_7$

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

Another Example

Comparing the previous results with the group presented by $\langle (a,\ b) \ | \ a^7=b^3=1,\ a^2=bab^{-1} \rangle$

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

Semidirect product using group action

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

Characters Table

Displaying characters table for the group $C_7 \rtimes C_3$ in $\textbf{S}_7$

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

Polynomial with Galois group

Computing Galois Group of irreductible polynomial $P=X^7-8X^5-2X^4+16X^3+6X^2-6X-2$ from GroupName website

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

Galois Theory

Galois Group of polynomial $P = X^5 + X^4 - 4X^3 - 3X^2 + 3X + 1$

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 $\bf{Gal}(\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q})=\mathbf{C_2}\times\mathbf{C_2}$

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 $\alpha^4-2 = 0,\ \bf{Gal}(\mathrm{Q}(\alpha, \mathrm{i})/\mathrm{Q}) = \mathbf{D_4}$

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

Illustration

Polynomial P=X^2-5 with Galois Group C5x:C4

References

ALGÈBRE T1 Daniel Guin, Thomas Hausberger. ALGÈBRE T1 Groupes, corps et théorie de Galois. EDP Sciences. All Chapters.

Algebra (3rd ed.) Saunders MacLane, Garrett Birkhoff. Algebra (3rd ed.). American Mathematical Society. Chapter XII.2 Groups extensions.

A Course on Finite Groups H.E. Rose. A Course on Finite Groups. Springer. 2009. All Chapters. Web Sections, Web Chapters.

A Course on Finite Groups H.E. Rose. A Course on Finite Groups. Web Sections, Web Chapters and Solution Appendix. Springer. 2009. Chapter 13. Representation and Character Theory.

GroupNames Tim Dokchitser. Group Names. Beta

GAP2022 The GAP Group. GAP -- Groups, Algorithms, and Programming. Version 4.12.0; 2022.

Paper (pdf) Ken Brown. Mathematics 7350. Cornell University. The Todd–Coxeter procedure

Conway polynomials Frank Lübeck. Conway polynomials for finite fields Online data

AECF Alin Bostan, Frédéric Chyzak, Marc Giusti, Romain Lebreton, Grégoire Lecerf, Bruno Salvy, Éric Schost. Algorithmes Efficaces en Calcul Formel. Édition web 1.1, 2018 Chapitre IV Factorisation des polynômes.

hal-01444183 Xavier Caruso Computations with p-adic numbers. Journées Nationales de Calcul Formel, In press, Les cours du CIRM. 2017. Several implementations of p-adic numbers

Algebraic Factoring Barry Trager Algebraic Factoring and Rational Function Integration. Laboratory for Computer Science, MIT. 1976. Norms and Algebraic Factoring

Bases de Gröbner Jean-Charles Faugere Résolution des systèmes polynômiaux en utilisant les bases de Gröbner. INRIA (POLSYS) / UPMC / CNRS/ LIP6. (JNCF) 2015

Ideals, Varieties, and Algorithms David A. Cox, John Little, Donal O’Shea Ideals, Varieties, and Algorithms. Springer. 4th Edition. 2015. Chapter 5. Polynomial and Rational Functions on a Variety

Charaktertheorie Jun.-Prof. Dr. Caroline Lassueur Character Theory of Finite Groups. TU Kaiserslautern, SS2020. Chapter 6. Induction and Restriction of Characters

Wikipedia Wikipedia. The Free Encyclopedia. Many algorithms and proofs

WolframCloud Wolfram Research. WOLFRAM CLOUD Integrated Access to Computational Intelligence 2023 Wolfram Research, Inc.

Pari/GP The PARI Group. PARI/GP Number Theory-oriented Computer Algebra System. Bordeaux. Version 2.15.x; 2020-2022.

milneFT Milne, James S. Fields and Galois Theory. Ed web v5.10, 2022. Chapter 4. Computing Galois Groups.

ChatGPT OpenAI. ChatGPT August 3, 2023 Version3.5 Free Research Preview.

fplll The FPLLL development team. A lattice reduction library may2021 Version 5.4.1

arb F.Johansson Arb a C library for arbitrary-precision ball arithmetic june2022 Version 2.23

milneCFT Milne, James S. Class Field Theory. Ed web v4.03, 2020. Chapter 2. The Cohomology of Groups.

Cohomology of Finite Groups Alejandro Adem, R. James Milgram. Cohomology of Finite Groups Springer. Second Edition 2004. Chapter 1. Group Extensions, Simple Algebras and Cohomology

arXiv:math/0010134 Florentin Smarandache, PhD Associate Professor Chair of Department of Math & Sciences INTEGER ALGORITHMS TO SOLVE DIOPHANTINE LINEAR EQUATIONS AND SYSTEMS. University of New Mexico nov 2007.

RSW1999 Iain Raeburn, Aidan Sims, and Dana P.Williams TWISTED ACTIONS AND OBSTRUCTIONS IN GROUP COHOMOLOGY. Germany, March 8–12, 1999.

A000001 OEIS. The On-Line Encyclopedia of Integer Sequences. A000001 Number of groups of order n.

relators John J. CANNON. Construction of defining relators for finite groups. Department of Pure Mathematics, University of Sydney. Discrete Mathematics 5 (1973), North-Holland Publishing Company. Construction of defining relators for finite groups. p105-129

generatorsGLnq D.E.Taylor. Pairs of Generators for Matrix Groups. Department of Pure Mathematics The University of Sydney Australia 2006. arXiv:2201.09155v1

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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...

finite-groupsabelian-groupsquotient-groupspermutationssemi-direct-productcsharphomomorphismsautomorphism-groupgalois-grouppolynomial-factoringfinite-fieldstodd-coxeterfield-extensionsgrobner-basischaracter-tablefield-theorygroup-theorymultiprecisionbch-codescohomology

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