public void GapPerm_Compose_Identity()
{
var f = new GapPerm();
var g = new GapPerm(0, 2, 1);
Assert.AreEqual(g.Compose(f), g);
}
static void Main(string[] args)
{
// Find the order of each of the following permutations
int proc(GapPerm a)
{
var result = new GapPerm();
foreach (var n in Enumerable.Range(1, 100))
{
result = result.Compose(a);
WriteLine("a^{0} = {1}", n, result);
if (result == new GapPerm(0))
{
return(n);
}
}
throw new Exception();
}
proc(new GapPerm("(12)(345)")).Display(); WriteLine();
proc(new GapPerm("(12)(3456)")).Display(); WriteLine();
proc(new GapPerm("(1234)(56789)")).Display(); WriteLine();
}
public void GapPerm_Identity_Compose()
{
var f = new GapPerm();
var g = new GapPerm(0, 2, 1);
Assert.AreEqual(f.Compose(g), g);
}
static void Main(string[] args)
{
var ε = new GapPerm(0);
var α = new GapPerm("(12)(34)(56)");
var β = new GapPerm("(23)");
var αβ = α.Compose(β);
var βα = β.Compose(α);
var αβα = α.Compose(β.Compose(α));
var βαβ = β.Compose(α.Compose(β));
var αβαβ = α.Compose(β.Compose(α.Compose(β)));
string lookup(GapPerm f)
{
var items = new[] {
(ε, "ε"), (α, "α"), (β, "β"),
(αβ, "αβ"), (βα, "βα"), (αβα, "αβα"), (βαβ, "βαβ"), (αβαβ, "αβαβ")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var G = new Group <GapPerm>
{
Identity = ε,
Set = new MathSet <GapPerm>(new[] { ε, α, β, αβ, βα, αβα, βαβ, αβαβ }),
Op = (a, b) => a.Compose(b),
Lookup = lookup
};
foreach (var elt in G.Set)
{
WriteLine("{0,-4}: {1}", lookup(elt), elt);
}
WriteLine();
G.ShowInverses();
G.ShowOperationTableColored(); WriteLine();
MathSet <int> orbit(Group <GapPerm> grp, int u) => grp.Set.ConvertAll(elt => elt.Apply(u));
WriteLine("orbit of 1: {0}", orbit(G, 1));
WriteLine("orbit of 2: {0}", orbit(G, 2));
WriteLine("orbit of 3: {0}", orbit(G, 3));
WriteLine("orbit of 5: {0}", orbit(G, 5));
WriteLine();
MathSet <GapPerm> stabilizer(Group <GapPerm> grp, int u) =>
grp.Set.Where(elt => elt.Apply(u) == u).ToMathSet();
WriteLine("stabilizer of 1: {0}", stabilizer(G, 1).Select(lookup).ToMathSet());
WriteLine("stabilizer of 2: {0}", stabilizer(G, 2).Select(lookup).ToMathSet());
WriteLine("stabilizer of 4: {0}", stabilizer(G, 4).Select(lookup).ToMathSet());
WriteLine("stabilizer of 5: {0}", stabilizer(G, 5).Select(lookup).ToMathSet());
}
static int order(GapPerm a)
{
var result = new GapPerm();
foreach (var n in Enumerable.Range(1, 100))
{
result = result.Compose(a);
if (result == new GapPerm(0))
{
return(n);
}
}
throw new Exception();
}
static void Main(string[] args)
{
Console.OutputEncoding = Encoding.UTF8;
var ε = new GapPerm(0); // new FunctionIntInt((1, 1), (2, 2), (3, 3));
var α = new GapPerm("(23)"); // new FunctionIntInt((1, 1), (2, 3), (3, 2));
var β = new GapPerm("(132)"); // new FunctionIntInt((1, 3), (2, 1), (3, 2));
var γ = new GapPerm("(12)"); // new FunctionIntInt((1, 2), (2, 1), (3, 3));
var σ = new GapPerm("(123)"); // new FunctionIntInt((1, 2), (2, 3), (3, 1));
var κ = new GapPerm("(13)"); // new FunctionIntInt((1, 3), (2, 2), (3, 1));
string lookup(GapPerm f)
{
var items = new[] { (ε, "ε"), (α, "α"), (β, "β"), (γ, "γ"), (σ, "σ"), (κ, "κ") };
return(items.First(elt => f == elt.Item1).Item2);
}
var S3 = new Group <GapPerm>
{
Identity = ε,
Set = new[] { ε, α, β, γ, σ, κ }.ToMathSet(),
Op = (a, b) => a.Compose(b),
Lookup = lookup
};
// var S3 = new Group(new[] { ε, α, β, γ, σ, κ }, (a, b) => a.Compose(b), lookup)
Write("S3 "); S3.ShowOperationTableColored(); WriteLine();
// consider - G.IsSubgroup(elts)
var H = S3.Subgroup(new[] { ε, β, σ });
WriteLine($"H: { H.Set.ConvertAll(lookup) }\n");
foreach (var elt in S3.CosetGrouping(H, "H"))
{
WriteLine("{0} {1}", elt.ToMathSet(), elt.Key.ConvertAll(lookup));
}
WriteLine();
Write("Z10/{ ε, β, σ } ");
S3.QuotientGroup(H, coset => new[] { ε, α, β, γ, σ, κ }.ToList().IndexOf(coset.Element).ToString()).ShowOperationTableColored();
}
static void Main(string[] args)
{
var R0 = new GapPerm(0); // R0
var R1 = new GapPerm("(123456)"); // R60
var R2 = new GapPerm("(135)(246)"); // R120
var R3 = new GapPerm("(14)(25)(36)"); // R180
var R4 = new GapPerm("(153)(264)"); // R240
var R5 = new GapPerm("(165432)"); // R300
var R6 = new GapPerm("(15)(24)"); // FA
var R7 = new GapPerm("(16)(25)(34)"); // FB
var R8 = new GapPerm("(26)(35)"); // FC
var R9 = new GapPerm("(12)(36)(45)"); // FD
var R10 = new GapPerm("(13)(46)"); // FE
var R11 = new GapPerm("(14)(23)(56)"); // FF
string lookup(GapPerm f)
{
var items = new[] {
(R0, "R0"),
(R1, "R1"),
(R2, "R2"),
(R3, "R3"),
(R4, "R4"),
(R5, "R5"),
(R6, "R6"),
(R7, "R7"),
(R8, "R8"),
(R9, "R9"),
(R10, "R10"),
(R11, "R11")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var D6 = new Group <GapPerm>
{
Identity = R0,
Set = new MathSet <GapPerm>(new[] { R0, R1, R2, R3, R4, R5, R6, R7, R8, R9, R10, R11 }),
Op = (a, b) => a.Compose(b),
Lookup = lookup
};
D6.ShowOperationTableColored();
}
static void Main(string[] args)
{
// A C B
// \ | /
// 1---2
// | | - D
// 4---3
var R0 = new GapPerm(0); // R0
var R1 = new GapPerm("(1234)"); // R90
var R2 = new GapPerm("(13)(24)"); // R180
var R3 = new GapPerm("(4321)"); // R270
var R4 = new GapPerm("(24)"); // FA
var R5 = new GapPerm("(13)"); // FB
var R6 = new GapPerm("(12)(34)"); // FC
var R7 = new GapPerm("(14)(23)"); // FD
string lookup(GapPerm f)
{
var items = new[] {
(R0, "R0"),
(R1, "R1"),
(R2, "R2"),
(R3, "R3"),
(R4, "R4"),
(R5, "R5"),
(R6, "R6"),
(R7, "R7")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var D4 = new Group <GapPerm>
{
Identity = R0,
Set = new MathSet <GapPerm>(new[] { R0, R1, R2, R3, R4, R5, R6, R7 }),
Op = (a, b) => a.Compose(b),
Lookup = lookup
};
WriteLine("subgroups : {0}", D4.Subgroups());
WriteLine("normal subgroups: {0}", D4.NormalSubgroups());
}
static void Main(string[] args)
{
{
var permutation = new GapPerm(0, 4, 9, 2, 5, 1, 7, 6, 8, 3);
permutation.DisplayAsFunction();
permutation.ToDisjointCycles().Display();
WriteLine();
}
{
var permutation = new GapPerm(0, 7, 4, 9, 2, 3, 8, 1, 6, 5);
permutation.DisplayAsFunction();
permutation.ToDisjointCycles().Display();
WriteLine();
}
{
var permutation = new GapPerm(0, 7, 9, 5, 3, 1, 2, 4, 8, 6);
permutation.DisplayAsFunction();
permutation.ToDisjointCycles().Display();
WriteLine();
}
{
var permutation = new GapPerm(0, 9, 8, 7, 4, 3, 6, 5, 1, 2);
permutation.DisplayAsFunction();
permutation.ToDisjointCycles().Display();
WriteLine();
}
}
static void Main(string[] args)
{
var perm = new GapPerm("(123)");
{
var f = new GapPerm("(23)");
var g = new GapPerm("(56)");
var h = f.Compose(g);
}
{
var f = new GapPerm("(234)");
var g = new GapPerm("(345)");
var h = f.Compose(g);
}
{
var f = new GapPerm("(12)(34)");
}
{
var f = new GapPerm("(123)");
var result = new[]
{
f.Apply(0),
f.Apply(1),
f.Apply(3),
f.Apply(4)
};
}
{
var f = new GapPerm("(12)");
var g = new GapPerm(0, 2, 1);
WriteLine(f.Equals(g));
WriteLine(f == g);
WriteLine(f.GetHashCode());
WriteLine(g.GetHashCode());
}
{
var f = new GapPerm("(1234)");
var result = f.Inverse();
}
{
var f = new GapPerm(10, 20, 30, 40, 4).Simplify();
}
{
var f = new GapPerm();
var g = new GapPerm(0, 2, 1);
var result_a = f.Apply(10);
var result_b = f.Inverse();
var result_c = f.Compose(g);
var result_d = g.Compose(f);
// var result_e = f.
}
{
var f = new GapPerm(3, 1, 2, 0);
var g = f.Inverse();
}
{
var f = new GapPerm(0, 2, 1, 3);
var g = f.Inverse();
}
{
var f = new GapPerm("(123)(234)(456)").ToDisjointCycles();
var g = new GapPerm("(12)(3456)").ToDisjointCycles();
var h = new GapPerm("(123)(234)(456)");
}
}
public static GapPerm κ = new GapPerm("(13)"); // new FunctionIntInt((1, 3), (2, 2), (3, 1));
static string lookup(GapPerm f)
{
var items = new[] { (ε, "ε"), (α, "α"), (β, "β"), (γ, "γ"), (σ, "σ"), (κ, "κ") };
static void Main(string[] args)
{
// A C B
// \ | /
// 1---2
// | | - D
// 4---3
var R0 = new GapPerm(0); // R0
var R1 = new GapPerm("(1234)"); // R90
var R2 = new GapPerm("(13)(24)"); // R180
var R3 = new GapPerm("(4321)"); // R270
var R4 = new GapPerm("(24)"); // FA
var R5 = new GapPerm("(13)"); // FB
var R6 = new GapPerm("(12)(34)"); // FC
var R7 = new GapPerm("(14)(23)"); // FD
string lookup(GapPerm f)
{
var items = new[] {
(R0, "R0"),
(R1, "R1"),
(R2, "R2"),
(R3, "R3"),
(R4, "R4"),
(R5, "R5"),
(R6, "R6"),
(R7, "R7")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var D4 = new Group <GapPerm>
{
Identity = R0,
Set = new MathSet <GapPerm>(new[] { R0, R1, R2, R3, R4, R5, R6, R7 }),
Op = (a, b) => a.Compose(b),
Lookup = lookup,
OpString = "·"
};
Write("D4 "); D4.ShowOperationTableColored(); WriteLine();
var H = D4.Subgroup(new[] { R0, R2, R4, R5 });
WriteLine("Elements of H: {0}\n", H.Set.ConvertAll(H.Lookup));
WriteLine("Elements of quotient group D4/H:\n");
foreach (var elt in D4.CosetGrouping(H, "H"))
{
WriteLine($"{ elt.ToMathSet() } { elt.Key.ConvertAll(lookup) }");
}
WriteLine();
Write("D4/{R0 R2 R4 R5} ");
D4.QuotientGroup(H).ShowOperationTableColored();
}
static void Main(string[] args)
{
// A C B
// \ | /
// 1---2
// | | - D
// 4---3
var R0 = new GapPerm(0); // R0
var R1 = new GapPerm("(1234)"); // R90
var R2 = new GapPerm("(13)(24)"); // R180
var R3 = new GapPerm("(4321)"); // R270
var R4 = new GapPerm("(24)"); // FA
var R5 = new GapPerm("(13)"); // FB
var R6 = new GapPerm("(12)(34)"); // FC
var R7 = new GapPerm("(14)(23)"); // FD
string lookup(GapPerm f)
{
var items = new[] {
(R0, "R0"),
(R1, "R1"),
(R2, "R2"),
(R3, "R3"),
(R4, "R4"),
(R5, "R5"),
(R6, "R6"),
(R7, "R7")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var D4 = new Group <GapPerm>
{
Identity = R0,
Set = new MathSet <GapPerm>(new[] { R0, R1, R2, R3, R4, R5, R6, R7 }),
Op = (a, b) => a.Compose(b),
Lookup = lookup,
OpString = "·"
};
Write("D4 "); D4.ShowOperationTableColored(); WriteLine();
foreach (var a in D4.Set)
{
foreach (var b in D4.Set)
{
WriteLine("{0}·{1}·{2}·{3} -> {4}",
lookup(a),
lookup(b),
lookup(D4.Inverse(a)),
lookup(D4.Inverse(b)),
lookup(D4.Op_(a, b, D4.Inverse(a), D4.Inverse(b))));
}
WriteLine();
}
WriteLine();
void ShowCommutators <T>(Group <T> G)
{
WriteLine("commutators:\n");
var width = G.Set.Select(elt => G.Lookup(elt).Count()).Max();
var color_thunks = new List <Action>()
{
() => { ForegroundColor = ConsoleColor.White; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Red; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Yellow; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Green; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Blue; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Cyan; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Magenta; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkRed; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkBlue; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkGreen; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkMagenta; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkYellow; BackgroundColor = ConsoleColor.Black; }
};
void pick_color(T elt)
{
var i = G.Set.ToList().IndexOf(elt);
color_thunks[i % color_thunks.Count()]();
}
Write("{0}|", "".PadLeft(width));
foreach (var elt in G.Set)
{
pick_color(elt); Write("{0} ", G.Lookup(elt).PadLeft(width));
}
WriteLine();
color_thunks[0](); WriteLine(new String('-', (G.Set.Count() + 1) * (width + 1)));
foreach (var x in G.Set)
{
pick_color(x); Write("{0}", G.Lookup(x).PadLeft(width)); color_thunks[0](); Write("|");
foreach (var y in G.Set)
{
// var elt = G.Op(x, y);
var elt = G.Op_(x, y, G.Inverse(x), G.Inverse(y));
pick_color(elt);
Write("{0} ", G.Lookup(elt).PadLeft(width));
}
WriteLine();
}
ForegroundColor = ConsoleColor.White;
BackgroundColor = ConsoleColor.Black;
}
ShowCommutators(D4); WriteLine();
var H = D4.Subgroup(new[] { R0, R2 });
foreach (var elt in D4.CosetGrouping(H, "H"))
{
WriteLine("{0} {1}",
String.Join(" ", elt.Select(item => String.Format("{0} =", item.ToString()))),
elt.Key.ConvertAll(D4.Lookup));
}
WriteLine();
Write("D4/H "); D4.QuotientGroup(H, "H").ShowOperationTableColored();
}
static void Main(string[] args)
{
Console.OutputEncoding = Encoding.UTF8;
//MathSet<int> center(Group<int> G)
//{
// return G.Set.Where(a => G.Set.All(b => G.Op(a, b) == G.Op(b, a))).ToMathSet();
//}
// G.Set.Where(elt => G.Set.All(b => Commute(a,b)))
// G.Set.Where(elt => G.Set.All(CommutesWith(a)))
//MathSet<GapPerm> center_GapPerm(Group<GapPerm> G)
//{
// return G.Set.Where(a => G.Set.All(b => G.Op(a, b) == G.Op(b, a))).ToMathSet();
//}
var group_table_gap_perm = new List <(Group <GapPerm>, string)>();
var group_table_int = new List <(Group <int>, string)>();
var group_table_mathset_int = new List <(Group <MathSet <int> >, string)>();
{
var ε = new GapPerm(0); // new FunctionIntInt((1, 1), (2, 2), (3, 3));
var α = new GapPerm("(23)"); // new FunctionIntInt((1, 1), (2, 3), (3, 2));
var β = new GapPerm("(132)"); // new FunctionIntInt((1, 3), (2, 1), (3, 2));
var γ = new GapPerm("(12)"); // new FunctionIntInt((1, 2), (2, 1), (3, 3));
var σ = new GapPerm("(123)"); // new FunctionIntInt((1, 2), (2, 3), (3, 1));
var κ = new GapPerm("(13)"); // new FunctionIntInt((1, 3), (2, 2), (3, 1));
string lookup(GapPerm f)
{
var items = new[] { (ε, "ε"), (α, "α"), (β, "β"), (γ, "γ"), (σ, "σ"), (κ, "κ") };
return(items.First(elt => f == elt.Item1).Item2);
}
var S3 = new Group <GapPerm>
{
Identity = ε,
Set = new[] { ε, α, β, γ, σ, κ }.ToMathSet(),
Op = (a, b) => a.Compose(b),
Lookup = lookup
};
// (S3, "S3")
// WriteLine(center_GapPerm(S3));
WriteLine("S3 center : {0}", S3.Center());
group_table_gap_perm.Add((S3, "S3"));
}
{
// A C B
// \ | /
// 1---2
// | | - D
// 4---3
var R0 = new GapPerm(0); // R0
var R1 = new GapPerm("(1234)"); // R90
var R2 = new GapPerm("(13)(24)"); // R180
var R3 = new GapPerm("(4321)"); // R270
var R4 = new GapPerm("(24)"); // FA
var R5 = new GapPerm("(13)"); // FB
var R6 = new GapPerm("(12)(34)"); // FC
var R7 = new GapPerm("(14)(23)"); // FD
string lookup(GapPerm f)
{
var items = new[] {
(R0, "R0"),
(R1, "R1"),
(R2, "R2"),
(R3, "R3"),
(R4, "R4"),
(R5, "R5"),
(R6, "R6"),
(R7, "R7")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var D4 = new Group <GapPerm>
{
Identity = R0,
Set = new MathSet <GapPerm>(new[] { R0, R1, R2, R3, R4, R5, R6, R7 }),
Op = (a, b) => a.Compose(b),
Lookup = lookup,
OpString = "·"
};
Write("D4 "); D4.ShowOperationTableColored();
WriteLine("D4 center : {0}\n", D4.Center());
// D4/C
// D4.QuotientGroup(D4.Center())
{
var C = D4.Subgroup(D4.Center());
Write("D4/C ");
D4.QuotientGroup(C).ShowOperationTableColored();
}
WriteLine();
group_table_gap_perm.Add((D4, "D4"));
}
{
var R0 = new GapPerm(0); // R0
var R1 = new GapPerm("(123456)"); // R60
var R2 = new GapPerm("(135)(246)"); // R120
var R3 = new GapPerm("(14)(25)(36)"); // R180
var R4 = new GapPerm("(153)(264)"); // R240
var R5 = new GapPerm("(165432)"); // R300
var R6 = new GapPerm("(15)(24)"); // FA
var R7 = new GapPerm("(16)(25)(34)"); // FB
var R8 = new GapPerm("(26)(35)"); // FC
var R9 = new GapPerm("(12)(36)(45)"); // FD
var R10 = new GapPerm("(13)(46)"); // FE
var R11 = new GapPerm("(14)(23)(56)"); // FF
string lookup(GapPerm f)
{
var items = new[] {
(R0, "R0"),
(R1, "R1"),
(R2, "R2"),
(R3, "R3"),
(R4, "R4"),
(R5, "R5"),
(R6, "R6"),
(R7, "R7"),
(R8, "R8"),
(R9, "R9"),
(R10, "R10"),
(R11, "R11")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var D6 = new Group <GapPerm>
{
Identity = R0,
Set = new MathSet <GapPerm>(new[] { R0, R1, R2, R3, R4, R5, R6, R7, R8, R9, R10, R11 }),
Op = (a, b) => a.Compose(b),
Lookup = lookup,
OpString = "·"
};
Write("D6 "); D6.ShowOperationTableColored(); WriteLine();
WriteLine("D6 center : {0}", D6.Center());
// D6/C
{
var C = D6.Subgroup(D6.Center());
Write("D6/C ");
D6.QuotientGroup(C).ShowOperationTableColored();
}
WriteLine();
group_table_gap_perm.Add((D6, "D6"));
}
{
var Z8 = new Group <int>
{
Identity = 0,
Set = Enumerable.Range(0, 8).ToMathSet(),
Op = (a, b) => (a + b) % 8,
OpString = "+"
};
WriteLine("Z8 center : {0}", Z8.Center());
group_table_int.Add((Z8, "Z8"));
}
{
var Z10 = new Group <int>
{
Identity = 0,
Set = Enumerable.Range(0, 10).ToMathSet(),
Op = (a, b) => (a + b) % 10,
OpString = "+"
};
WriteLine("Z10 center : {0}", Z10.Center());
group_table_int.Add((Z10, "Z10"));
}
{
IEnumerable <T> SymmetricDifference <T>(IEnumerable <T> a, IEnumerable <T> b) =>
a.Except(b).Union(b.Except(a));
var P3 = new Group <MathSet <int> >
{
Identity = new MathSet <int> {
},
Set = new[] { 1, 2, 3 }.PowerSet().Select(elt => elt.ToMathSet()).ToMathSet(),
Op = (a, b) => SymmetricDifference(a, b).ToMathSet()
};
WriteLine("P3 center : {0}", P3.Center());
group_table_mathset_int.Add((P3, "P3"));
}
{
var a = new List <int> {
10, 20, 30
};
var b = new List <string> {
"abc", "bcd", "cde"
};
var c = new List <double> {
1.2, 2.3, 3.4
};
var ls = new List <System.Collections.IList> {
a, b, c
};
//foreach (var elt in ls)
// Console.WriteLine(elt.Count);
// var ls = new List<List<object>> { a, b, c };
// Group interface - IGroup
// new List<IGroup>
}
// S3 D4 D6 Z8 Z10 P3
foreach (var tup in group_table_gap_perm)
{
WriteLine("{0,-4} size : {1,3} center size : {2,3}", tup.Item2, tup.Item1.Set.Count, tup.Item1.Center().Count);
}
foreach (var tup in group_table_int)
{
WriteLine("{0,-4} size : {1,3} center size : {2,3}", tup.Item2, tup.Item1.Set.Count, tup.Item1.Center().Count);
}
foreach (var tup in group_table_mathset_int)
{
WriteLine("{0,-4} size : {1,3} center size : {2,3}", tup.Item2, tup.Item1.Set.Count, tup.Item1.Center().Count);
}
}
public static GapPerm R7 = new GapPerm("(14)(23)"); // FD
static string lookup(GapPerm f)
{
var items = new[] {
static void Main(string[] args)
{
var R0 = new GapPerm(0); // R0
var R1 = new GapPerm("(123456)"); // R60
var R2 = new GapPerm("(135)(246)"); // R120
var R3 = new GapPerm("(14)(25)(36)"); // R180
var R4 = new GapPerm("(153)(264)"); // R240
var R5 = new GapPerm("(165432)"); // R300
var R6 = new GapPerm("(15)(24)"); // FA
var R7 = new GapPerm("(16)(25)(34)"); // FB
var R8 = new GapPerm("(26)(35)"); // FC
var R9 = new GapPerm("(12)(36)(45)"); // FD
var R10 = new GapPerm("(13)(46)"); // FE
var R11 = new GapPerm("(14)(23)(56)"); // FF
string lookup(GapPerm f)
{
var items = new[] {
(R0, "R0"),
(R1, "R1"),
(R2, "R2"),
(R3, "R3"),
(R4, "R4"),
(R5, "R5"),
(R6, "R6"),
(R7, "R7"),
(R8, "R8"),
(R9, "R9"),
(R10, "R10"),
(R11, "R11")
};
return(items.First(elt => f == elt.Item1).Item2);
}
var D6 = new Group <GapPerm>
{
Identity = R0,
Set = new MathSet <GapPerm>(new[] { R0, R1, R2, R3, R4, R5, R6, R7, R8, R9, R10, R11 }),
Op = (a, b) => a.Compose(b),
Lookup = lookup,
OpString = "·"
};
Write("D6 "); D6.ShowOperationTableColored(); WriteLine();
foreach (var a in D6.Set)
{
foreach (var b in D6.Set)
{
WriteLine("{0,3}·{1,3}·{2,3}·{3,3} -> {4,3}",
lookup(a),
lookup(b),
lookup(D6.Inverse(a)),
lookup(D6.Inverse(b)),
lookup(D6.Op_(a, b, D6.Inverse(a), D6.Inverse(b))));
}
WriteLine();
}
WriteLine();
void ShowCommutators <T>(Group <T> G)
{
WriteLine("commutators:\n");
var width = G.Set.Select(elt => G.Lookup(elt).Count()).Max();
var color_thunks = new List <Action>()
{
() => { ForegroundColor = ConsoleColor.White; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Red; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Yellow; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Green; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Blue; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Cyan; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.Magenta; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkRed; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkBlue; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkGreen; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkMagenta; BackgroundColor = ConsoleColor.Black; },
() => { ForegroundColor = ConsoleColor.DarkYellow; BackgroundColor = ConsoleColor.Black; }
};
void pick_color(T elt)
{
var i = G.Set.ToList().IndexOf(elt);
color_thunks[i % color_thunks.Count()]();
}
Write("{0}|", "".PadLeft(width));
foreach (var elt in G.Set)
{
pick_color(elt); Write("{0} ", G.Lookup(elt).PadLeft(width));
}
WriteLine();
color_thunks[0](); WriteLine(new String('-', (G.Set.Count() + 1) * (width + 1)));
foreach (var x in G.Set)
{
pick_color(x); Write("{0}", G.Lookup(x).PadLeft(width)); color_thunks[0](); Write("|");
foreach (var y in G.Set)
{
// var elt = G.Op(x, y);
var elt = G.Op_(x, y, G.Inverse(x), G.Inverse(y));
pick_color(elt);
Write("{0} ", G.Lookup(elt).PadLeft(width));
}
WriteLine();
}
ForegroundColor = ConsoleColor.White;
BackgroundColor = ConsoleColor.Black;
}
Write("D6 "); ShowCommutators(D6); WriteLine();
var H = D6.Subgroup(new[] { R0, R2, R4 }); // D6 is size 12. H is size 3. 12/3 -> 4 cosets
foreach (var elt in D6.CosetGrouping(H, "H"))
{
WriteLine("{0} {1}",
String.Join(" ", elt.Select(item => String.Format("{0} =", item.ToString()))),
elt.Key.ConvertAll(D6.Lookup));
}
WriteLine();
Write("D6/H "); D6.QuotientGroup(H, "H").ShowOperationTableColored();
}
static string even_or_odd(GapPerm obj) => obj.ToDisjointCycles().to_transpositions().ls.Count % 2 == 0 ? "even" : "odd";
