aidevnn / FineGrain

Finite group study, abelians or not, quotient group, direct product and many more...

FineGrain

Finite group study, abelians or not, quotient group, direct and semidirect product and many more...

Starting to study Quotient Group in Z.

var zn = new Group256(20);
var G = zn.Generate("Z", 1);
var H = zn.Generate("5Z", 5);
G.DisplayHead();
H.DisplayHead();

var Qg = new QuotientGroup<byte, Integer256>(G, H);
Qg.Details();
Qg.DisplayClasses();

Will output

|Z| = 20
IsGroup      : True
IsCommutative: True

|5Z| = 4
IsGroup      : True
IsCommutative: True

|Z/5Z| = 5 with |Z| = 20 and |5Z| = 4, OpBoth
IsGroup      : True
IsCommutative: True

@ = (  0)[  1]
a = (  1)[ 20]
b = (  2)[ 10]
c = (  3)[ 20]
d = (  4)[  5]

|Z/5Z| = 5 with |Z| = 20 and |5Z| = 4, OpBoth
 *|@ a b c d
--|----------
 @|@ a b c d
 a|a b c d @
 b|b c d @ a
 c|c d @ a b
 d|d @ a b c


Class of : (  0)[  1]
    Represent
    (  0)[  1]
    (  5)[  4]
    ( 10)[  2]
    ( 15)[  4]
Class of : (  1)[ 20]
    Represent
    (  1)[ 20]
    (  6)[ 10]
    ( 11)[ 20]
    ( 16)[  5]
Class of : (  2)[ 10]
    Represent
    (  2)[ 10]
    (  7)[ 20]
    ( 12)[  5]
    ( 17)[ 20]
Class of : (  3)[ 20]
    Represent
    (  3)[ 20]
    (  8)[  5]
    ( 13)[ 20]
    ( 18)[ 10]
Class of : (  4)[  5]
    Represent
    (  4)[  5]
    (  9)[ 20]
    ( 14)[ 10]
    ( 19)[ 20]

Simple S3, the group of permutations

var sn = new Sigma(3);
var H0 = sn.Generate("H0", (1, 2, 3), (1, 2));
var H1 = sn.Generate("H1", (1, 2, 3));
H0.DisplayElements();
H1.DisplayElements();

var Qg0 = new QuotientGroup<byte, Permutation>(H0, H1);
Qg0.Details();
Qg0.DisplayClasses();

Will output

|H0| = 6
IsGroup      : True
IsCommutative:False

@ = ( 1  2  3)[ 1+]
a = ( 1  3  2)[ 2-]
b = ( 2  1  3)[ 2-]
c = ( 3  2  1)[ 2-]
d = ( 2  3  1)[ 3+]
e = ( 3  1  2)[ 3+]

|H1| = 3
IsGroup      : True
IsCommutative: True

@ = ( 1  2  3)[ 1+]
a = ( 2  3  1)[ 3+]
b = ( 3  1  2)[ 3+]

|H0/H1| = 2 with |H0| = 6 and |H1| = 3, OpBoth
IsGroup      : True
IsCommutative: True

@ = ( 1  2  3)[ 1+]
a = ( 1  3  2)[ 2-]

|H0/H1| = 2 with |H0| = 6 and |H1| = 3, OpBoth
 *|@ a
--|----
 @|@ a
 a|a @


Class of : ( 1  2  3)[ 1+]
    Represent
    ( 1  2  3)[ 1+]
    ( 2  3  1)[ 3+]
    ( 3  1  2)[ 3+]
Class of : ( 1  3  2)[ 2-]
    Represent
    ( 1  3  2)[ 2-]
    ( 2  1  3)[ 2-]
    ( 3  2  1)[ 2-]

Then S5

var H0 = sn.Generate("H0", (1, 2, 3), (4, 5));
var H1 = sn.Generate("H1", (4, 5));
H0.DisplayElements();
H1.DisplayElements();

var Qg0 = new QuotientGroup<byte, Permutation>(H0, H1);
Qg0.Details();
Qg0.DisplayClasses();

will output

|H0| = 6
IsGroup      : True
IsCommutative: True

@ = ( 1  2  3  4  5)[ 1+]
a = ( 1  2  3  5  4)[ 2-]
b = ( 2  3  1  4  5)[ 3+]
c = ( 3  1  2  4  5)[ 3+]
d = ( 2  3  1  5  4)[ 6-]
e = ( 3  1  2  5  4)[ 6-]

|H1| = 2
IsGroup      : True
IsCommutative: True

@ = ( 1  2  3  4  5)[ 1+]
a = ( 1  2  3  5  4)[ 2-]

|H0/H1| = 3 with |H0| = 6 and |H1| = 2, OpBoth
IsGroup      : True
IsCommutative: True

@ = ( 1  2  3  4  5)[ 1+]
a = ( 2  3  1  4  5)[ 3+]
b = ( 3  1  2  4  5)[ 3+]

|H0/H1| = 3 with |H0| = 6 and |H1| = 2, OpBoth
 *|@ a b
--|------
 @|@ a b
 a|a b @
 b|b @ a


Class of : ( 1  2  3  4  5)[ 1+]
    Represent
    ( 1  2  3  4  5)[ 1+]
    ( 1  2  3  5  4)[ 2-]
Class of : ( 2  3  1  4  5)[ 3+]
    Represent
    ( 2  3  1  4  5)[ 3+]
    ( 2  3  1  5  4)[ 6-]
Class of : ( 3  1  2  4  5)[ 3+]
    Represent
    ( 3  1  2  4  5)[ 3+]
    ( 3  1  2  5  4)[ 6-]