[iojs][python][sympy]Alternating Group(A4, A5) multiplication table (generator)

table-a4.js

"use strict";

var A4 = {
    E: [0,1,2,3], I: [1,0,3,2], J: [2,3,0,1], K: [3,2,1,0],
    S: [2,0,1,3], T: [],U: [],V: [],
    W: [1,2,0,3], X: [],Y: [],Z: [],
};

var mul = function(a, b) {
    return [a[b[0]], a[b[1]], a[b[2]], a[b[3]]];
};
A4.T = mul(A4.S, A4.I);A4.U = mul(A4.S, A4.J);A4.V = mul(A4.S, A4.K);
A4.X = mul(A4.W, A4.I);A4.Y = mul(A4.W, A4.J);A4.Z = mul(A4.W, A4.K);

var name = function (g) {
    for (let k of Object.keys(A4)) {
        let h = A4[k];
        if (g[0] === h[0] && g[1] === h[1] && g[2] === h[2] && g[3] === h[3])
            return k;
    }
    throw Error("not found: " + g);
};

var keys = Object.keys(A4);
var table = keys.map(function (a) {
    return keys.map(function (b) {
        return name(mul(A4[a], A4[b]));
    });
});

var sym = ["α", "β", "γ", "δ"];
keys.forEach(function (k) {
    let syms = A4[k].map(function (i) {return sym[i];});
    console.log(k + "=(" + syms.join(",") + ")");
});

console.log(" |" + keys.join("|") + "|");
table.forEach(function (line, i) {
    console.log(keys[i] + "|" + line.join("|") + "|");
});

table-a4.py

import sympy
import sympy.combinatorics as symg

a4e = symg.Permutation(3)        # identity of (0,1,2,3) permutation
a4i = symg.Permutation(0,1)(2,3) # 2-swap 0=>1, 1=>0, 2=>3, 3=>2
a4s = symg.Permutation(3)(0,1,2) # rot-3 as 0=>1, 1=>2, 2=>0, 3=>3

a4 = symg.PermutationGroup(a4i, a4s) # == AlternatingGrpup(4)
assert a4.order() == 12

#print(list(a4.generate())) # list of 12 permutations
assert list(a4.generate())[0] == a4e

h4 = a4.derived_subgroup() # == DihedralGroup(2)
i4 = h4.derived_subgroup() # == trivial 4
assert h4.order() == 4
assert h4.is_normal(a4)
assert i4.is_trivial

# example of props of group
a3 = symg.AlternatingGroup(3)
a5 = symg.AlternatingGroup(5)
assert a3.is_primitive()
assert not a4.is_primitive()
assert a5.is_primitive()
assert a3.is_solvable
assert a4.is_solvable
assert not a5.is_solvable

# print permutations in a4
h4l = list(h4.generate())
assert h4l[0] == a4e

from collections import OrderedDict
names = OrderedDict(sorted({
    a4e: "E",
    h4l[1]: "I",
    h4l[2]: "J",
    h4l[3]: "K",
    a4s: "S",
    a4s * h4l[1]: "T",
    a4s * h4l[2]: "U",
    a4s * h4l[3]: "V",
    a4s * a4s: "W",
    a4s * a4s * h4l[1]: "X",
    a4s * a4s * h4l[2]: "Y",
    a4s * a4s * h4l[3]: "Z",
}.items(), key=lambda t: t[1]))
assert len(names) == 12
#print(names)

for p, n in names.items():
    print("{0}=({1})".format(n, ",".join("αβγδ"[i] for i in p.array_form)))
    pass
print()


def chunk(n, it):
    return list(zip(*([iter(it)] * n))) # trick from grouper() in itertools doc

fmt = "{}||" + (("{}" * 4) + "|") * 3
print(fmt.format("*", *names.values()))
print(fmt.format("=", *(["="] * len(names))))
for ch in chunk(4, names.items()):
    for p, n in ch:
        print(fmt.format(n, *(names[p * q] for q in names.keys())))
        pass
    print(fmt.format("-", *(["-"] * len(names))))
    pass

table-a4.txt

E=(α,β,γ,δ)
I=(β,α,δ,γ)
J=(γ,δ,α,β)
K=(δ,γ,β,α)
S=(γ,α,β,δ)
T=(α,γ,δ,β)
U=(β,δ,γ,α)
V=(δ,β,α,γ)
W=(β,γ,α,δ)
X=(γ,β,δ,α)
Y=(α,δ,β,γ)
Z=(δ,α,γ,β)
 |E|I|J|K|S|T|U|V|W|X|Y|Z|
E|E|I|J|K|S|T|U|V|W|X|Y|Z|
I|I|E|K|J|V|U|T|S|Y|Z|W|X|
J|J|K|E|I|T|S|V|U|Z|Y|X|W|
K|K|J|I|E|U|V|S|T|X|W|Z|Y|
S|S|T|U|V|W|X|Y|Z|E|I|J|K|
T|T|S|V|U|Z|Y|X|W|J|K|E|I|
U|U|V|S|T|X|W|Z|Y|K|J|I|E|
V|V|U|T|S|Y|Z|W|X|I|E|K|J|
W|W|X|Y|Z|E|I|J|K|S|T|U|V|
X|X|W|Z|Y|K|J|I|E|U|V|S|T|
Y|Y|Z|W|X|I|E|K|J|V|U|T|S|
Z|Z|Y|X|W|J|K|E|I|T|S|V|U|

table-a4py.txt

E=(α,β,γ,δ)
I=(β,α,δ,γ)
J=(γ,δ,α,β)
K=(δ,γ,β,α)
S=(β,γ,α,δ)
T=(α,δ,β,γ)
U=(δ,α,γ,β)
V=(γ,β,δ,α)
W=(γ,α,β,δ)
X=(δ,β,α,γ)
Y=(α,γ,δ,β)
Z=(β,δ,γ,α)

*||EIJK|STUV|WXYZ|
=||====|====|====|
E||EIJK|STUV|WXYZ|
I||IEKJ|VUTS|YZWX|
J||JKEI|TSVU|ZYXW|
K||KJIE|UVST|XWZY|
-||----|----|----|
S||STUV|WXYZ|EIJK|
T||TSVU|ZYXW|JKEI|
U||UVST|XWZY|KJIE|
V||VUTS|YZWX|IEKJ|
-||----|----|----|
W||WXYZ|EIJK|STUV|
X||XWZY|KJIE|UVST|
Y||YZWX|IEKJ|VUTS|
Z||ZYXW|JKEI|TSVU|
-||----|----|----|

table-a5.js

"use strict";

var permutations = function (l) {
    if (l.length === 1) return [l.slice(0, 1)];
    let ps = permutations(l.slice(1));
    let r = [];
    ps.forEach(function (p) {
        for (let i = 0; i < l.length; i++) {
            r.push(p.slice(0, i).concat(l.slice(0, 1)).concat(p.slice(i)));
        }
    });
    return r;
};
var even = function (g) {
    let n = 0;
    for (let i = 0; i < g.length - 1; i++) {
        for (let j = i + 1; j < g.length; j++) {
            if (g[i] > g[j]) n++;
        }
    }
    return n % 2 === 0;
};

var s5 = permutations([0,1,2,3,4]);
var a5 = s5.filter(even);
var mul = function (a, b) {
    return b.map(function (k) {return a[k];});
};
//console.log(a5);
var a5names = "0123456789" + "ABCDEFGHIJ" + "KLMNOPQRST" + "UVWXYabcde" +
    "fghijklmno" + "pqrstuvwxy";
var name = function (g) {
    for (let i = 0; i < a5.length; i++) {
        let h = a5[i];
        if (g.every(function (k, j) {return k === h[j];})) return a5names[i];
    }
};
//a5.forEach(function (g) {console.log(name(g));});

var table = a5.map(function (g) {
    return a5.map(function (h) {
        return name(mul(g, h));
    });
});

var sym = ["α", "β", "γ", "δ", "ε"];
a5.forEach(function (g, i) {
    let syms = g.map(function (i) {return sym[i];});
    console.log(a5names[i] + "=(" + syms.join(",") + ")");
});
console.log();

console.log(" |" + a5names + "|");
console.log("-|" + "-".repeat(a5.length) + "|");
table.forEach(function (line, i) {
    console.log(a5names[i] + "|" + line.join("") + "|");
});
console.log("-|" + "-".repeat(a5.length) + "|");


// cultering with conjugacy class
var a5inv = a5.map(function (g) {
    for (var i = 0; i < a5.length; i++) {
        if (name(mul(g, a5[i])) === "0") return a5[i];
    }
});
var conj = function (h) {
    var gen = a5.map(function (g, i) {
        var gm = a5inv[i];
        return name(mul(g, mul(h, gm)));
    });
    return gen.filter(function (n, i) {
        return gen.indexOf(n) === i;
    }).sort();
};
var classes = function (names) {
    return names.map(function (n) {
        return a5[a5names.indexOf(n)];
    });
};
console.log();

console.log("<<conjugacy classes>>");
console.log("unit:", conj(a5[0]).join("")); // 1 elem
console.log("3-rots:", conj(a5[1]).join("")); // 20 elems
console.log("2-swaps:", conj(a5[6]).join("")); // 15 elems
console.log("5-rots by 12340:", conj(a5[2]).join(""));  // 12 elems 
console.log("5-rots by 23140:", conj(a5[7]).join(""));  // 12 elems


console.log();
console.log("<<reorderd elements with conjugacy classes>>");
//console.log("5-rots by 12340:", classes(conj(a5[2])));  // 12 elems 
//console.log("5-rots by 23140:", classes(conj(a5[7])));  // 12 elems

var u = classes(conj(a5[0]));
var r3 = classes(conj(a5[1]));
var s2 = classes(conj(a5[6]));
var r5a = classes(conj(a5[2]));
var r5b = classes(conj(a5[7]));

var cls = [u, r3, s2, r5a, r5b];
var clstable = cls.map(function (cla) {
    return cla.map(function (a) {
        return cls.map(function (clb) {
            return clb.map(function (b) {
                return mul(a, b);
            });
        });
    });
});

console.log();
var i = 0;
var clsa5 = [];
cls.forEach(function (cl) {
    console.log("-".repeat(20));
    cl.forEach(function (g) {
        clsa5.push(g);
        let syms = g.map(function (i) {return sym[i];});
        console.log(a5names[i] + "=(" + syms.join(",") + ")");
        i++;
    });
});
var cname = function (g) {
    for (var i = 0; i < clsa5.length; i++) {
        let h = clsa5[i];
        if (g.every(function (k, j) {return k === h[j];})) return a5names[i];
    }
};

console.log();
console.log(" |" + cls.map(function (cl) {
    return cl.map(cname).join("");
}).join("|") + "|");
var i = 0;
clstable.forEach(function (cla) {
    console.log("-+" + cls.map(function (cl) {
        return cl.map(function () {return "-"}).join("");
    }).join("+") + "+");

    cla.forEach(function (mcl) {
        console.log(a5names[i] + "|" + mcl.map(function (ms) {
            return ms.map(cname).join("");
        }).join("|") + "|");
        i++;
    });
});
console.log("-+" + cls.map(function (cl) {
    return cl.map(function () {return "-"}).join("");
}).join("+") + "+");

table-a5.txt

0=(α,β,γ,δ,ε)
1=(β,γ,α,δ,ε)
2=(β,γ,δ,ε,α)
3=(γ,α,β,δ,ε)
4=(γ,β,δ,α,ε)
5=(α,γ,δ,β,ε)
6=(γ,δ,α,β,ε)
7=(γ,δ,β,ε,α)
8=(γ,α,δ,ε,β)
9=(γ,δ,ε,α,β)
A=(β,α,δ,γ,ε)
B=(β,δ,γ,α,ε)
C=(α,δ,β,γ,ε)
D=(δ,β,α,γ,ε)
E=(δ,β,γ,ε,α)
F=(δ,α,γ,β,ε)
G=(δ,γ,β,α,ε)
H=(α,δ,γ,ε,β)
I=(δ,γ,α,ε,β)
J=(δ,γ,ε,β,α)
K=(α,β,δ,ε,γ)
L=(β,δ,α,ε,γ)
M=(β,δ,ε,γ,α)
N=(δ,α,β,ε,γ)
O=(δ,β,ε,α,γ)
P=(α,δ,ε,β,γ)
Q=(δ,ε,α,β,γ)
R=(δ,ε,β,γ,α)
S=(δ,α,ε,γ,β)
T=(δ,ε,γ,α,β)
U=(β,α,γ,ε,δ)
V=(β,γ,ε,α,δ)
W=(α,γ,β,ε,δ)
X=(γ,β,α,ε,δ)
Y=(γ,β,ε,δ,α)
a=(γ,α,ε,β,δ)
b=(γ,ε,β,α,δ)
c=(α,γ,ε,δ,β)
d=(γ,ε,α,δ,β)
e=(γ,ε,δ,β,α)
f=(α,β,ε,γ,δ)
g=(β,ε,α,γ,δ)
h=(β,ε,γ,δ,α)
i=(ε,α,β,γ,δ)
j=(ε,β,γ,α,δ)
k=(α,ε,γ,β,δ)
l=(ε,γ,α,β,δ)
m=(ε,γ,β,δ,α)
n=(ε,α,γ,δ,β)
o=(ε,γ,δ,α,β)
p=(β,α,ε,δ,γ)
q=(β,ε,δ,α,γ)
r=(α,ε,β,δ,γ)
s=(ε,β,α,δ,γ)
t=(ε,β,δ,γ,α)
u=(ε,α,δ,β,γ)
v=(ε,δ,β,α,γ)
w=(α,ε,δ,γ,β)
x=(ε,δ,α,γ,β)
y=(ε,δ,γ,β,α)

 |0123456789ABCDEFGHIJKLMNOPQRSTUVWXYabcdefghijklmnopqrstuvwxy|
-|------------------------------------------------------------|
0|0123456789ABCDEFGHIJKLMNOPQRSTUVWXYabcdefghijklmnopqrstuvwxy|
1|13805ACHKP46BGIDFLNS279EJMRTOQXaUWcfkprwVbdjlginsuYehmotyqvx|
2|279EJMRTOQYehmotyqvxVbdjlginsu46BGIDFLNS13805ACHKPXaUWcfkprw|
3|30K1A4BL2M5C6FNGD7EO8HPIS9TQJRWfXUpVgYhqakrlibjsmtcwdnuoxeyv|
4|4BLGD6FNIS1A30K5C8HPXUpWfakrcw2M7EOJR9TQYhqmteyvoxVgbjslidnu|
5|567FGCDENO34012ABKLMWXYUVfghpq89HIJSTPQRcdenowxyuvabklmijrst|
6|6FN5C30KWfGD4BL1AXUp7EO2MYhqVgIS8HPcwakr9TQoxdnuliJReyvmtbjs|
7|7EO2MYhqVgJReyvmtbjs9TQoxdnuliGD4BL1AXUp6FN5C30KWfIS8HPcwakr|
8|8HPIS9TQJRcwdnuoxeyvakrlibjsmt5C6FNGD7EO30K1A4BL2MWfXUpVgYhq|
9|9TQoxdnuliIS8HPcwakr6FN5C30KWfJReyvmtbjs7EO2MYhqVgGD4BL1AXUp|
A|ACHDFBGIEJ0513846279UWcXaVbdYeKPLNSOQMRTprwsuqvxtyfkginjlhmo|
B|BGI46138XaDFACH05UWcLNSKPprwfkEJ279YeVbdMRTtyhmojlOQqvxsugin|
C|CDEAB012UVFG56734WXYHIJ89cdeabNOKLMpqfghPQRuvrstijSTwxynoklm|
D|DAUC0F5WHcB1G4X63I8aE2V7YJeb9dLpNKfPrSwkOqgvsQuixnMhRtjymTol|
E|E2V7YJeb9dMhRtjymTolOqgvsQuixnB1G4X63I8aDAUC0F5WHcLpNKfPrSwk|
F|F5W63G4X7YC0DAUB1E2VNKfLpOqgMhHcI8a9dJebSwkxnTolymPrQuivsRtj|
G|G4XB1DAULp63F5WC0NKfI8aHcSwkPr7YE2VMhOqgJebymRtjvs9dTolxnQui|
H|HIJ89cdeabSTwxynoklmPQRuvrstijFG56734WXYCDEAB012UVNOKLMpqfgh|
I|I8aHcSwkPr9dTolxnQuiJebymRtjvs63F5WC0NKfG4XB1DAULp7YE2VMhOqg|
J|JebymRtjvs7YE2VMhOqgG4XB1DAULp9dTolxnQuiI8aHcSwkPr63F5WC0NKf|
K|KLMNOPQRSTpqrstuvwxyfghijklmnoABCDEFGHIJ0123456789UVWXYabcde|
L|LNSKPprwfkOQqvxsuginMRTtyhmojlDFACH05UWcBGI46138XaEJ279YeVbd|
M|MRTtyhmojlEJ279YeVbdBGI46138XaOQqvxsuginLNSKPprwfkDFACH05UWc|
N|NKfLpOqgMhPrQuivsRtjSwkxnTolymC0DAUB1E2VF5W63G4X7YHcI8a9dJeb|
O|OqgvsQuixnLpNKfPrSwkDAUC0F5WHcMhRtjymTolE2V7YJeb9dB1G4X63I8a|
P|PQRuvrstijNOKLMpqfghCDEAB012UVSTwxynoklmHIJ89cdeabFG56734WXY|
Q|QuiPrNKfC0vsOqgLpDAURtjMhE2VB1xnSwkHcF5WTol9dI8a63ymJeb7YG4X|
R|RtjMhE2VB1ymJeb7YG4XTol9dI8a63vsOqgLpDAUQuiPrNKfC0xnSwkHcF5W|
S|SwkxnTolymHcI8a9dJebF5W63G4X7YPrQuivsRtjNKfLpOqgMhC0DAUB1E2V|
T|Tol9dI8a63xnSwkHcF5WQuiPrNKfC0ymJeb7YG4XRtjMhE2VB1vsOqgLpDAU|
U|UWcXaVbdYefkginjlhmoprwsuqvxty0513846279ACHDFBGIEJKPLNSOQMRT|
V|VbdjlginsuXaUWcfkprw13805ACHKPYehmotyqvx279EJMRTOQ46BGIDFLNS|
W|WXYUVfghpqabklmijrstcdenowxyuv34012ABKLM567FGCDENO89HIJSTPQR|
X|XUpWfakrcwVgbjslidnuYhqmteyvox1A30K5C8HP4BLGD6FNIS2M7EOJR9TQ|
Y|Yhqmteyvox2M7EOJR9TQ4BLGD6FNISVgbjslidnuXUpWfakrcw1A30K5C8HP|
a|akrlibjsmtWfXUpVgYhq30K1A4BL2Mcwdnuoxeyv8HPIS9TQJR5C6FNGD7EO|
b|bjsVgXUp1AliakrWf30Kdnucw8HP5CmtYhq2M4BLeyvJR7EOGDox9TQIS6FN|
c|cdenowxyuv89HIJSTPQR567FGCDENOabklmijrstWXYUVfghpq34012ABKLM|
d|dnucw8HP5Cox9TQIS6FNeyvJR7EOGDliakrWf30KbjsVgXUp1AmtYhq2M4BL|
e|eyvJR7EOGDmtYhq2M4BLbjsVgXUp1Aox9TQIS6FNdnucw8HP5CliakrWf30K|
f|fghijklmnoUVWXYabcde0123456789pqrstuvwxyKLMNOPQRSTABCDEFGHIJ|
g|ginfkUWc05jlVbdXa138hmoYe27946suprwKPACHqvxOQLNSDFtyMRTEJBGI|
h|hmoYe27946tyMRTEJBGIqvxOQLNSDFjlVbdXa138ginfkUWc05suprwKPACH|
i|if0gUjV1h2kWla3bXmY4nc5d8o96e7rKspAqLtMBuPCQNvODREwHxSFTIyJG|
j|jV1bXla3d8gUif0kWnc5spArKuPCwHh2mY4e7o96tMBREyJGTIqLvODQNxSF|
k|klmabWXY34ijfghUV012rstpqKLMABnocde89567wxySTHIJFGuvPQRNOCDE|
l|la3kWif0rKbXjV1gUspAmY4h2tMBqLd8nc5wHuPCo96TIxSFQNe7yJGREvOD|
m|mY4h2tMBqLe7yJGREvODo96TIxSFQNbXjV1gUspAla3kWif0rKd8nc5wHuPC|
n|nc5d8o96e7wHxSFTIyJGuPCQNvODREkWla3bXmY4if0gUjV1h2rKspAqLtMB|
o|o96TIxSFQNd8nc5wHuPCla3kWif0rKe7yJGREvODmY4h2tMBqLbXjV1gUspA|
p|prwsuqvxtyKPLNSOQMRTACHDFBGIEJfkginjlhmoUWcXaVbdYe0513846279|
q|qvxOQLNSDFsuprwKPACHginfkUWc05tyMRTEJBGIhmoYe27946jlVbdXa138|
r|rstpqKLMABuvPQRNOCDEwxySTHIJFGijfghUV012klmabWXY34nocde89567|
s|spArKuPCwHqLvODQNxSFtMBREyJGTIgUif0kWnc5jV1bXla3d8h2mY4e7o96|
t|tMBREyJGTIh2mY4e7o96jV1bXla3d8qLvODQNxSFspArKuPCwHgUif0kWnc5|
u|uPCQNvODRErKspAqLtMBif0gUjV1h2wHxSFTIyJGnc5d8o96e7kWla3bXmY4|
v|vODqLspAgUQNuPCrKif0xSFwHnc5kWREtMBh2jV1yJGe7mY4bXTIo96d8la3|
w|wxySTHIJFGnocde89567klmabWXY34uvPQRNOCDErstpqKLMABijfghUV012|
x|xSFwHnc5kWTIo96d8la3yJGe7mY4bXQNuPCrKif0vODqLspAgUREtMBh2jV1|
y|yJGe7mY4bXREtMBh2jV1vODqLspAgUTIo96d8la3xSFwHnc5kWQNuPCrKif0|
-|------------------------------------------------------------|

<<conjugacy classes>>
unit: 0
3-rots: 1345BCDEFHKYcfhjknrs
2-swaps: 6AGOPTUWXdmptwy
5-rots by 12340: 2IJLSabeiqvx
5-rots by 23140: 789MNQRVglou

<<reorderd elements with conjugacy classes>>

--------------------
0=(α,β,γ,δ,ε)
--------------------
1=(β,γ,α,δ,ε)
2=(γ,α,β,δ,ε)
3=(γ,β,δ,α,ε)
4=(α,γ,δ,β,ε)
5=(β,δ,γ,α,ε)
6=(α,δ,β,γ,ε)
7=(δ,β,α,γ,ε)
8=(δ,β,γ,ε,α)
9=(δ,α,γ,β,ε)
A=(α,δ,γ,ε,β)
B=(α,β,δ,ε,γ)
C=(γ,β,ε,δ,α)
D=(α,γ,ε,δ,β)
E=(α,β,ε,γ,δ)
F=(β,ε,γ,δ,α)
G=(ε,β,γ,α,δ)
H=(α,ε,γ,β,δ)
I=(ε,α,γ,δ,β)
J=(α,ε,β,δ,γ)
K=(ε,β,α,δ,γ)
--------------------
L=(γ,δ,α,β,ε)
M=(β,α,δ,γ,ε)
N=(δ,γ,β,α,ε)
O=(δ,β,ε,α,γ)
P=(α,δ,ε,β,γ)
Q=(δ,ε,γ,α,β)
R=(β,α,γ,ε,δ)
S=(α,γ,β,ε,δ)
T=(γ,β,α,ε,δ)
U=(γ,ε,α,δ,β)
V=(ε,γ,β,δ,α)
W=(β,α,ε,δ,γ)
X=(ε,β,δ,γ,α)
Y=(α,ε,δ,γ,β)
a=(ε,δ,γ,β,α)
--------------------
b=(β,γ,δ,ε,α)
c=(δ,γ,α,ε,β)
d=(δ,γ,ε,β,α)
e=(β,δ,α,ε,γ)
f=(δ,α,ε,γ,β)
g=(γ,α,ε,β,δ)
h=(γ,ε,β,α,δ)
i=(γ,ε,δ,β,α)
j=(ε,α,β,γ,δ)
k=(β,ε,δ,α,γ)
l=(ε,δ,β,α,γ)
m=(ε,δ,α,γ,β)
--------------------
n=(γ,δ,β,ε,α)
o=(γ,α,δ,ε,β)
p=(γ,δ,ε,α,β)
q=(β,δ,ε,γ,α)
r=(δ,α,β,ε,γ)
s=(δ,ε,α,β,γ)
t=(δ,ε,β,γ,α)
u=(β,γ,ε,α,δ)
v=(β,ε,α,γ,δ)
w=(ε,γ,α,β,δ)
x=(ε,γ,δ,α,β)
y=(ε,α,δ,β,γ)

 |0|123456789ABCDEFGHIJK|LMNOPQRSTUVWXYa|bcdefghijklm|nopqrstuvwxy|
-+-+--------------------+---------------+------------+------------+
0|0|123456789ABCDEFGHIJK|LMNOPQRSTUVWXYa|bcdefghijklm|nopqrstuvwxy|
-+-+--------------------+---------------+------------+------------+
1|1|204ML5Nc7ebDWuUwvKFV|639dqsTRSJICxkm|orfnOEHYGial|ABPp8tQghjyX|
2|2|01M36L9rNnoWCgJjhVUI|547fptSTRFKDyil|B8OAduvkwYma|ebqPcQsEHGXx|
3|3|5N7LM20B4oTOpCkXixhG|916EgYbn8QluKUy|eAPRDdtsVvjI|rcfWSHJqFamw|
4|4|L9N6301bMBSdPDixYyHw|725uEkoAcsagVJX|neqTWfQtIhGK|8rOCRvFpUmlj|
5|5|N3L19M6A0RepuqQaFGkl|274PWH8bnhxOmvI|cSDrECiUXsyj|oTgfBJYdtVwK|
6|6|7M50N4Ln2SAqEPtlJjYm|193pDhrBevXfaHV|8TCcgWkFyQxw|bRudoUiOsKGI|
7|7|M6091N3TLc8EfOvKsmtX|452CdUerBYjqGQw|RogbpPJHlFVx|SADunihWkyIa|
8|8|bnCdFtXGaQOTc7R09ArB|iqVKsI5N3oSeEfH|uxwkmL2g6WJY|hpUvlyj1M4DP|
9|9|4L2N07MR58rgdfHIQasy|361WOFAcoiwPjtG|SbuBqpUhmJKX|TnCEekvDYxVl|
A|A|copDQYmaIHPnS6850RBe|UfxlJG94LTbrqEF|dwVsj23CMOkv|ightyKXN71uW|
B|B|erOPkJKXyYE8A0b34oST|sWlGHxM67cnRCDi|qmavI9Nd2uhU|tfQFjwV51Lpg|
C|C|FVXiqn8Odp3KUTWEgD20|abt7LfuhGIJ1BoP|kQs5cwjySM6A|lxmeN9rvRHY4|
D|D|UIxYpAcdfP4VJSCuEW01|moQN6OgHwKF2bBq|istLrjGXR35e|ayln978hTvkM|
E|E|vjGHuSTCgD0XYBqOPf67|wRh34pWJKmtM8Ad|FUi1oylar5Nc|VIxb2LnkesQ9|
F|F|VCibaqtQ85kU1vIHR0WJ|nXdse9Guh2DKYMA|xNcl7TgoEyP6|p3LmOrfwjS4B|
G|G|uhTwRjE0HIK3xX58aQlO|gvSByAFVCpNk7m9|1D4WYinLterf|2UoMJP6bqdcs|
H|H|wghSGEvFR0Ji4YaQA9Ps|TjukB5IDULdyt68|V1bKMopnflO7|C23XWeqxmcNr|
I|I|DUoxAmf9Qay2Vj0RGFKW|pYcrl8HwgC1JMX5|4dNPthT3vBeq|Lin6sO7SEubk|
J|J|KWkBlPstr6YF0HVhS2DU|eyOQANjEv1CIi4n|X78m9RubgxpL|qM5afcdGwT3o|
K|K|WJByelO7smX0IG1TwUVC|Pkr8acvjED2F3xL|Mf9qQHS4hbnp|6YA5tdNRugoi|
-+-+--------------------+---------------+------------+------------+
L|L|9462735e1TnPgpsmUwia|0NMqCvcoAHydlhK|rRW8uDYJxtXG|BSEObFkfQIjV|
M|M|6795412o3bRfqWYykXvj|N0LguiBertmEIFx|AnpSCOsQKHwV|c8dDThUPJlaG|
N|N|3517294S6rcuOdhVtlQx|ML0DfJn8bkGpwsj|TBEoPqFvaUIy|ReWgAYHCiXKm|
O|O|klKsWrBEPf7GQ8uCdpN3|yeJ09DqtXxh5Tcg|vYHMAaVwn12o|jmIR64SFbiUL|
P|P|sylJOBeqWE6aHAdpDg4L|Krk50ufYmwi9nSC|tvF7RIxVoN3T|XjG8M1bQcUh2|
Q|Q|xpUcIfYHA9shNtGF85Ok|omDJr0adi3ulv7R|w4Sy6nCTqKWM|gL2jPBEVXb1e|
R|R|STguHvjIGFWobMA958er|hEwyka012ncBfqQ|DVxJX3Lp7Pst|UCiYKlm46NdO|
S|S|TRuEhHwVjJDbB4nN6rAc|vgGxYl201e8odPt|CKXUyM5q9pQs|FWkiIma3L7Of|
T|T|RSEgvhGKwUCBo3e7Lcn8|HujXim120ArbOps|WIyFx46PNqtQ|JDYkValM59fd|
U|U|IDYompQscLiJ2hKvT1CF|Axftn7wgH0WVk3e|y9raNSEBuXq5|P46ld8OjGRMb|
V|V|CFbXnadNtlx1Kw2SjJID|qi8cmrhGuW0U4y6|3O7psvRMHoAP|5keLQf9TgEBY|
W|W|JKykPerfOqMIFRDguC12|lBs95dEvjVU0obp|YtQ68GwxT4Ln|mXaA7NcHShi3|
X|X|qt8abVC3ixG7mKMByYjE|dFnTwoklOf6v0I4|5pLuUsr9JRSD|NQc1hg2eWPAH|
Y|Y|mfQAxDUio4Ht6JXkBMEv|cIphS3yPs7qjF0b|aLnw2rO8WGu1|d9NVgTClKe5R|
a|a|dinV8Xq5FGlLwm9AIHyP|CtbeKRQxpg4s6j0|Nu1OvUo2YrBE|3hT7kWMcfDSJ|
-+-+--------------------+---------------+------------+------------+
b|b|n8dqiFVxXkuce1o4MBRS|tCawvy35NrATDWY|plmhK79f0gHJ|QOsUGjIL26PE|
c|c|oADfUQxwmsdSrNT17e8b|YpIVtKL94BRnuOv|gyjil60E5CFk|HPJhaXG23MWq|
d|d|iaVtC8buqONwscgDfP94|XnF17WpQxyHLSrE|hkv3emIjA20B|GlKT5MRUoYJ6|
e|e|rBPWsklmKvqAR5cL1Tbn|JOyaFw7M6So8puU|fjItG04D3dih|YEHQXVx9N2gC|
f|f|YmIQDcogpd9jtrEWOq7M|xAU2NCPsyXv6R8u|Hih4nlKGe01b|waVSL3TJBkF5|
g|g|HwjhETRWuC2yioPfpdL9|GSvM3qDUIas4rnO|JFk0bxmlc678|KVXB15eYAQtN|
h|h|GuvTjgHJS2Uk3iltnNpQ|RwEYo6VCF5OxsLr|K0BI4bqedmf9|W1MyDAPXa87c|
i|i|adtnXCFkb3hsLUyYo4gH|8VqvTMxpQ9PwJ2B|l5eG1cfrDjE0|ON7KuRWmIA6S|
j|j|EvRGSwg2hVIMXy6rltmf|uHToxnJKWq7Y9aN|0C3Dike5sAcd|1Fb4UpLBPO8Q|
k|k|lOseyWJYBMvQ5Fxib3uh|rKPHR4XqtNpGU1o|m6Aj08dcCwg2|f79IESDaVnLT|
l|l|OkeKryP6Jjm5GaNnVhxp|WsBAIStXqu3QLw2|7E0fHFb1icog|MvR9YD48dCTU|
m|m|fYAIcxpLUwa6jl7eKvXq|DQonVTsyPEMt5G1|9g2dhJB0k8bu|4HSNiC3rOWRF|
-+-+--------------------+---------------+------------+------------+
n|n|8bqCtialVhpeTLr62SoA|FdXmUjN35RBcPgJ|OGKQw1MW4fYH|kuvsxIy790ED|
o|o|AcfpYUIyxigrn2BM3bTR|QDmjhX4L98eSWCk|PalHVN7O1EvF|sdtJwGK605qu|
p|p|QxmUfoAPDgLlhnOqCu35|IcY62EdiaGkNeTW|sHJ9SVXKb7MR|ywjr40Bt8Fv1|
q|q|tXaFdbnpCu5mvefPWEM6|V8iL1gOkljY7ARD|QhUNTKyIB94S|xGwc32osrJH0|
r|r|BeWOJsyjltfR89S2Nnco|kPKIQV67MbTAgdh|EXGYa51uLDUi|vqFHmxw043Cp|
s|s|yPJrKOkve7tH9QwUcLdi|BlWF81mfY4gahNT|jMRX5ADSpVC3|E60GqbuIxo2n|
t|t|XqF8VdihnNQv7sjJr6fY|baCUc2lOkMEmH9S|G3TxLeWRPID4|u51wpogKyB0A|
u|u|hGwvgRSDEW1xkbpdqO5N|jTH4MPCFVlQ3cef|UJY2BXam8L9r|IKyo06Aints7|
v|v|jEHRwuhUT1FYMkmse7qt|SGgibLKWJ6fXQ5c|I2oV3BPAOadN|D04xCnpylr98|
w|w|gHSjTGu1vKV4yxLcmsad|EhRbXeUIDP9iNl7|2WMCkYA6Qn8O|0JB3Fq5opfrt|
x|x|pQcmoID4YywNlV3bXkGu|fUASjBiadO5h1KM|LP6gJt87FTRW|9sr2HE0nCqev|
y|y|PsrlBKWMkXj9aI4oxiwg|OJeRGbYmfdLH2V3|6q5EFQcNUSTC|7t80vu1ADpnh|
-+-+--------------------+---------------+------------+------------+

see: http://en.wikipedia.org/wiki/Alternating_group

60 members of A5

E=(0,1,2,3,4)
X0I=(0,2,1,4,3), X0J, X0K, ... X4I, X4J,X4K   -- (5x3 elems two distinct exchange)
T01R=(0,1,3,4,2), T01L=(0,1,4,2,3), T02R, T02L,  ...., T34L   -- (10 x 2 elems, rot 3)
Fabcd=(1,2,3,4,0), Facbd, ... Fdcba --  (4! elems, 4times-swap all)

By the conjugacy class of A5, all 3-rotations included to the same single class.
(A4 is not, 3-rots split two classes: STUV and WXYZ).

All other elements of A5(unit, 2-swaps, 5-rotation) is generated from a multiplication of some of two 3-rotations.

So, Normal Subgroup of A5 becomes A5 itself (or only unit). -- A5 is a Simple Group.

• http://en.wikipedia.org/wiki/Simple_group

Generic Solution Formula includes a Discriminant (deepest sqrt part).

Discriminant splits each solutions of a polynomial equation(if the discriminant is 0, some solutions become same value -- multiple roots).
So the "Generic" Solution Formula should keep the discriminant at least.

Discriminant of Quintic Equation(5-th degree polynomial) D = (α-β)(α-γ)(α-δ)(α-ε)(β-γ)(β-δ)(β-ε)(γ-δ)(γ-ε)(δ-ε) from its 5-solutions αβγδε.

The D only keeps even permutations. (odd permutation turn D to -D, means D=0).
The 5 elements of even permutations forms the Alternating Group A5.

---

Property of discriminant of Quartic Formula(4th degree polynomial) has A4, of Cubic Formula has A3.

A4 has a Normal Sub Group(EIJK). Its factor group forms A3 (-- A3 can expand to A4).
This means any Generic Quartic Equation build from (or break to) some Cubic Equation.
(or Quintic Formula embeds Cubic Formula).

A5 is a simple group which cannot break to smaller (even permutation) groups except unit.
It is a difference between generic Quintic Equation and generic Quartic Equation.

---

Another property of no generic quintic formula is that A5 is not cyclic group that means open roots simply.

Solutions of open roots (X^n = C) forms cyclic relationship.
All solutions rotate with multiplying n-th root of 1.