jonghough/permuj

J scripts for calculating permutation groups

PermuJ

Collection of J scripts for doing calculations with permutation groups.

Includes, or will hopefully include:

Creating permutation presentations of symmetric, alternating, dihedral, cyclic groups.

Functionality includes:

• finding conjugacy classes and cosets
• enumerating conjugacy classes
• finding subgroups
• commutators and derived subgroups
• involutions and centers
• quotients
• automorphisms
• constructing modulo-multiplication groups for given positive integer
• direct products

Finding normal subgroups and group structure, and finding quotient groups, and automorphisms.

Why J?

The J programming language is ideal for doing any kind of mathematics, and is ideal for looking into permutations and groups.

Usage

Need to have J installed. Best to install J 8.X ( http://www.jsoftware.com/stable.htm )

Example

NB. Create the symmetric group on 4 letters:

s4 =: Sym 4

NB. Create the dihedral group of order 8

d4 =. Dih 4

NB. Create direct product of S4 and D4:

s4d4 =: s4 dir_prod d4

NB. get the center of s4d4

cent =. center s4d4

Incidentally, the size of the center is 192 (i.e. 192 permutations make up the center of S4xD4). This can be seen by #cent

192

NB. Get the derived subgroup of s4d4

derived_subgroup s4d4

which gives

0 1 2 3 4 5 6 7

0 3 1 2 4 5 6 7

0 2 3 1 4 5 6 7

3 1 0 2 4 5 6 7

2 1 3 0 4 5 6 7

1 0 3 2 4 5 6 7

2 0 1 3 4 5 6 7

3 2 1 0 4 5 6 7

1 2 0 3 4 5 6 7

1 3 2 0 4 5 6 7

2 3 0 1 4 5 6 7

3 0 2 1 4 5 6 7

0 1 2 3 6 7 4 5

0 3 1 2 6 7 4 5

0 2 3 1 6 7 4 5

3 1 0 2 6 7 4 5

2 1 3 0 6 7 4 5

1 0 3 2 6 7 4 5

2 0 1 3 6 7 4 5

3 2 1 0 6 7 4 5

1 2 0 3 6 7 4 5

1 3 2 0 6 7 4 5

2 3 0 1 6 7 4 5

3 0 2 1 6 7 4 5

We might want to inspect the contents of the derived group:

dg =: derived_subgroup s4d4

conjugacy_classes dg

┌───────────────┬───────────────┬───────────────┬───────────────┬───────────────┬───────────────┬───────────────┬───────────────┐
│0 1 2 3 4 5 6 7│0 3 1 2 4 5 6 7│0 2 3 1 4 5 6 7│1 0 3 2 4 5 6 7│0 1 2 3 6 7 4 5│0 3 1 2 6 7 4 5│0 2 3 1 6 7 4 5│1 0 3 2 6 7 4 5│
│               │1 2 0 3 4 5 6 7│1 3 2 0 4 5 6 7│2 3 0 1 4 5 6 7│               │1 2 0 3 6 7 4 5│1 3 2 0 6 7 4 5│2 3 0 1 6 7 4 5│
│               │2 1 3 0 4 5 6 7│2 0 1 3 4 5 6 7│3 2 1 0 4 5 6 7│               │2 1 3 0 6 7 4 5│2 0 1 3 6 7 4 5│3 2 1 0 6 7 4 5│
│               │3 0 2 1 4 5 6 7│3 1 0 2 4 5 6 7│               │               │3 0 2 1 6 7 4 5│3 1 0 2 6 7 4 5│               │
└───────────────┴───────────────┴───────────────┴───────────────┴───────────────┴───────────────┴───────────────┴───────────────┘

The above gives a list of the conjugacy classes of dg.

We can also attempt to decompose dg into direct products.

decompose_directproduct dg

┌─────┬─────┬─────┐

│Alt 4│Sym 2│Sym 2│

└─────┴─────┴─────┘

Example: Calculate module multiplication group for the number 45:

mmg_groupnames 45

Example: Calculate possible sylow subgroups of S(5) (symmetirc group on 5 elements):

sylow3 Sym 5

Visualization

Visualizing the orbits of the elements of a group by way of the cayley table.

Below are visualizations of the cayley tables of Sym(5), Alt(5) and Dih(12), repsectively.

These are created by

 view_table Sym 5
 view_table Alt 5
 view_table Dih 12

It is also possible to view the orbit of a single element of the group. e.g. below is the orbit of a single element in the group Sym(3) x Cyc(5):

This is created by

1 view_orbit (Sym 3) dir_prod (Cyc 5) NB. left arg, 1, is the index of the permutation in Sym3xCyc5