Group

group is a small software intented to illustrate the theory of finite groups. This is a UNIX command (or DOS for Windows) which must be ran from any shell window. Give your operation table and group will test it for you and will return all subgroups with their properties. That's all! group is a free software distributed under the GNU general public licence. You can freely download, use, modify and re-distribute it.

A group is a set G in which an operation has been defined (noted * by convention). The couple (G,*) must verify the following axioms:
(A1) Closure: For any a,b in G a*b belongs to G
(A2) Associativity: For any a,b,c in G then (a*b)*c=a*(b*c)
(A3) Unity: It exists a unit element noted e such that e*a=a*e=a for any a in G
(A4) Inversibility: For any a in G, it exists an element noted a^-1 (the inverse of a) such that a*a^-1=a^-1*a=e

A group (G,*) is an Abelian group if the commutativity axiom is verified:
(A5) Commutativity: For any a,b in G then a*b=b*a

A subgroup H is a non-empty subset of G such that: For any a,b in H, a*b^-1 belongs to H
A subgroup is a group for the operation *

A subgroup H of G is a normal subgroup if: For any a in G and for any h in H, a*h*a^-1 belongs to H

Indeed, the mathematical theory of groups is much larger than the above summary. It exists many extended books and internet sites in which you can find all details and comments on the theory. But, we have limited the summary to the only definitions implemented in the software group.


Construct a group generated by two elements i and j such as

i²=e

j²=e

ij=ji

and find all subgroups and normal subgroups.

What do we have to do? The first step is to construct the operation table of the group by applying the algebraic relationships. We get,

The problem is now to check if axioms (A1) - (A5) apply. Indeed, for such an example, it can be done by hands. But you can imagine that the exercice becomes tedious (especially to check associativity) for larger groups. The other solution is to write a text file named for instance exercice.txt


# Input file to solve my exercice

ORDER=4

SET=e i j ij

OPERATION=+

TABLE=

       e  i  j ij

       i  e ij  j

       j ij  e  i

      ij  j  i  e

This is the imput file for group. It contains four fields which are,
- ORDER=n the number n of elements (order of the group)
- SET=a₁ ... a_n a list of symbols for elements (3 characters maximum per element)
- OPERATION=* a one character symbol (*, +, o or other) for the operation
- TABLE=a_i*a_j the operation table of the group. This is a table of n² entries where i and j runs from 1 to n.

Then run group by typing
group exercice.txt
in a command window. The output is,


Operation table

  *   e   i   j  ij 

  e   e   i   j  ij 

  i   i   e  ij   j 

  j   j  ij   e   i 

 ij  ij   j   i   e 



Inverse table

element   e   i   j  ij 

inverse   e   i   j  ij 



The set is an Abelian group



List of subgroups

{e} NORMAL

{e,i} NORMAL

{e,j} NORMAL

{e,ij} NORMAL

{e,i,j,ij} NORMAL

You can observe that all elements of the group are the product of elements of normal subgroups {e,i} and {e,j} which are cyclic of order 2 and whose intersection is {e}. The group is therefore Z/2Z X Z/2Z that is Klein's group.

History of releases

10 january 2010, release 0.1
- Check group axioms of any operation table.
- Find subgroups.
- Find normal subgroups.

Compiling group

To compile the source of group on Unix/Linux systems, download the tarball group-0.1.tar.gz and untar it with
tar -zxvf group-0.1.tar.gz
then enter in the directory group-0.1
cd group-0.1
configure the makefile with
./configure
compile the source
make
finally install the package (under root)
make install
It should well work. Complete instructions for installation are available in the file INSTALL.

Finite group software

What is group?

Goup theory

Example

Download

History of releases

Compiling group

Contact