FiniteGroup

Group theory. A simple C# code to show quickly generated group.

Sn.From((1, 2), (3, 4)).Details();

Will output the permutations (with its name, its order and its signature) and the generated group table.

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

|G| = 4 in S4
 *|@ a b c
--|--------
 @|@ a b c
 a|a @ c b
 b|b c @ a
 c|c b a @

And

Sn.From((1, 2), (1, 3)).Details();

Will output

|G| = 6 in S3
@ = ( 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+]

|G| = 6 in S3
 *|@ a b c d e
--|------------
 @|@ a b c d e
 a|a @ d e b c
 b|b e @ d c a
 c|c d e @ a b
 d|d c a b e @
 e|e b c a @ d

Disjunct 3-cycle and transposition.

Sn.From((2, 3, 1), (4, 5)).Details();

will produce a commutative group

|G| = 6 in S5
@ = ( 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-]

|G| = 6 in S5
 *|@ a b c d e
--|------------
 @|@ a b c d e
 a|a @ d e b c
 b|b d c @ e a
 c|c e @ b a d
 d|d b e a c @
 e|e c a d @ b

Z/nZ Groups are easiest to study

Zn.Dim(6).From(3).Details();
Zn.Dim(6).From(4).Details();
Zn.DetailZn(z6);

|G| = 2 in Z/6Z
@ = ( 0)[ 1]
a = ( 3)[ 2]

|G| = 2 in Z/6Z
 *|@ a
--|----
 @|@ a
 a|a @
 
|G| = 3 in Z/6Z
@ = ( 0)[ 1]
a = ( 2)[ 3]
b = ( 4)[ 3]

|G| = 3 in Z/6Z
 *|@ a b
--|------
 @|@ a b
 a|a b @
 b|b @ a
 
|G| = 6 in Z/6Z
@ = ( 0)[ 1]
a = ( 3)[ 2]
b = ( 2)[ 3]
c = ( 4)[ 3]
d = ( 1)[ 6]
e = ( 5)[ 6]

|G| = 6 in Z/6Z
 *|@ a b c d e
--|------------
 @|@ a b c d e
 a|a @ e d c b
 b|b e c @ a d
 c|c d @ b e a
 d|d c a e b @
 e|e b d a @ c

Direct Product of Z/nZ

Z/2Z isnt isomorphic to Z/4Z

Zn.Details(2, 2);

|G| = 4 in Z/2Z x Z/2Z
@ = ( 0,  0)[ 1]
a = ( 0,  1)[ 2]
b = ( 1,  0)[ 2]
c = ( 1,  1)[ 2]

|G| = 4 in Z/2Z x Z/2Z
 *|@ a b c
--|--------
 @|@ a b c
 a|a @ c b
 b|b c @ a
 c|c b a @

And

Zn.Details(4);

|G| = 4 in Z/4Z
@ = ( 0)[ 1]
a = ( 2)[ 2]
b = ( 1)[ 4]
c = ( 3)[ 4]

|G| = 4 in Z/4Z
 *|@ a b c
--|--------
 @|@ a b c
 a|a @ c b
 b|b c a @
 c|c b @ a
 

Comparing Z/2Z x Z/2Z x Z/2Z with Z/2Z x Z/4Z

Zn.Details(2, 2, 2);

|G| = 8 in Z/2Z x Z/2Z x Z/2Z
@ = ( 0,  0,  0)[ 1]
a = ( 0,  0,  1)[ 2]
b = ( 0,  1,  0)[ 2]
c = ( 0,  1,  1)[ 2]
d = ( 1,  0,  0)[ 2]
e = ( 1,  0,  1)[ 2]
f = ( 1,  1,  0)[ 2]
g = ( 1,  1,  1)[ 2]

|G| = 8 in Z/2Z x Z/2Z x Z/2Z
 *|@ a b c d e f g
--|----------------
 @|@ a b c d e f g
 a|a @ c b e d g f
 b|b c @ a f g d e
 c|c b a @ g f e d
 d|d e f g @ a b c
 e|e d g f a @ c b
 f|f g d e b c @ a
 g|g f e d c b a @

Zn.Details(2, 4);

|G| = 8 in Z/2Z x Z/4Z
@ = ( 0,  0)[ 1]
a = ( 0,  2)[ 2]
b = ( 1,  0)[ 2]
c = ( 1,  2)[ 2]
d = ( 0,  1)[ 4]
e = ( 0,  3)[ 4]
f = ( 1,  1)[ 4]
g = ( 1,  3)[ 4]

|G| = 8 in Z/2Z x Z/4Z
 *|@ a b c d e f g
--|----------------
 @|@ a b c d e f g
 a|a @ c b e d g f
 b|b c @ a f g d e
 c|c b a @ g f e d
 d|d e f g a @ c b
 e|e d g f @ a b c
 f|f g d e c b a @
 g|g f e d b c @ a

Comparing Z/12Z with Z/2Z x Z/6Z and Z/3Z x Z/4Z

Zn.Display(2, 6);
Zn.Display(3, 4);
Zn.Display(12);
            
|G| = 12 in Z/2Z x Z/6Z
@ = ( 0,  0)[ 1]
a = ( 0,  3)[ 2]
b = ( 1,  0)[ 2]
c = ( 1,  3)[ 2]
d = ( 0,  2)[ 3]
e = ( 0,  4)[ 3]
f = ( 0,  1)[ 6]
g = ( 0,  5)[ 6]
h = ( 1,  1)[ 6]
i = ( 1,  2)[ 6]
j = ( 1,  4)[ 6]
k = ( 1,  5)[ 6]

|G| = 12 in Z/3Z x Z/4Z
@ = ( 0,  0)[ 1]
a = ( 0,  2)[ 2]
b = ( 1,  0)[ 3]
c = ( 2,  0)[ 3]
d = ( 0,  1)[ 4]
e = ( 0,  3)[ 4]
f = ( 1,  2)[ 6]
g = ( 2,  2)[ 6]
h = ( 1,  1)[12]
i = ( 1,  3)[12]
j = ( 2,  1)[12]
k = ( 2,  3)[12]

|G| = 12 in Z/12Z
@ = ( 0)[ 1]
a = ( 6)[ 2]
b = ( 4)[ 3]
c = ( 8)[ 3]
d = ( 3)[ 4]
e = ( 9)[ 4]
f = ( 2)[ 6]
g = (10)[ 6]
h = ( 1)[12]
i = ( 5)[12]
j = ( 7)[12]
k = (11)[12]

Dihedral Dn generations

Sn.Dihedral(4);

(s * r) * (s * r) = id
s = ( 1  4  3  2)[ 2-]
r = ( 2  3  4  1)[ 4-]
s * r
  = ( 4  3  2  1)[ 2+]

|G| = 8 in S4
@ = ( 1  2  3  4)[ 1+]
a = ( 1  4  3  2)[ 2-]
b = ( 3  2  1  4)[ 2-]
c = ( 2  1  4  3)[ 2+]
d = ( 3  4  1  2)[ 2+]
e = ( 4  3  2  1)[ 2+]
f = ( 2  3  4  1)[ 4-]
g = ( 4  1  2  3)[ 4-]

|G| = 8 in S4
 *|@ a b c d e f g
--|----------------
 @|@ a b c d e f g
 a|a @ d f b g c e
 b|b d @ g a f e c
 c|c g f @ e d b a
 d|d b a e @ c g f
 e|e f g d c @ a b
 f|f e c a g b d @
 g|g c e b f a @ d

And D6

Sn.Dihedral(6);

(s * r) * (s * r) = id
s = ( 1  6  5  4  3  2)[ 2+]
r = ( 2  3  4  5  6  1)[ 6-]
s * r
  = ( 6  5  4  3  2  1)[ 2-]

|G| = 12 in S6
@ = ( 1  2  3  4  5  6)[ 1+]
a = ( 2  1  6  5  4  3)[ 2-]
b = ( 4  3  2  1  6  5)[ 2-]
c = ( 4  5  6  1  2  3)[ 2-]
d = ( 6  5  4  3  2  1)[ 2-]
e = ( 1  6  5  4  3  2)[ 2+]
f = ( 3  2  1  6  5  4)[ 2+]
g = ( 5  4  3  2  1  6)[ 2+]
h = ( 3  4  5  6  1  2)[ 3+]
i = ( 5  6  1  2  3  4)[ 3+]
j = ( 2  3  4  5  6  1)[ 6-]
k = ( 6  1  2  3  4  5)[ 6-]

|G| = 12 in S6
 *|@ a b c d e f g h i j k
--|------------------------
 @|@ a b c d e f g h i j k
 a|a @ h g i k j c b d f e
 b|b i @ e h c k j d a g f
 c|c g e @ f b d a k j i h
 d|d h i f @ j c k a b e g
 e|e j c b k @ h i f g a d
 f|f k j d c i @ h g e b a
 g|g c k a j h i @ e f d b
 h|h d a k b g e f i @ c j
 i|i b d j a f g e @ h k c
 j|j e f i g d a b c k h @
 k|k f g h e a b d j c @ i