Finite Group Software

group is a small software intended to illustrate the theory of finite groups. This is a UNIX command (or DOS for Windows) which must be run 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 License. You can freely download, use, modify and re-distribute it.

Definition of group

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:

There 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, there exists an element noted a^-1 (the inverse of a) such that a*a^-1 = a^-1*a = e

Abelian group

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

Subgroup

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 *

Normal subgroup

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. There exist 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.

Here is an example of use of group. In books of mathematics devoted to theory of groups, first exercises on groups are sometimes based on the explicit construction of groups by giving algebraic relationships on elements. For example,

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 hand. But you can imagine that the exercise becomes tedious (especially to check associativity) for larger groups.

The other solution is to write a text file named for instance exercice.txt

This is the input file for group. It contains four fields which are:

• ORDER=n the number n of elements (order of the group)
• SET=a1 ... an a list of symbols for elements (3 characters maximum per element)
• OPERATION=* a one character symbol (*, +, o or other) for the operation
• TABLE=ai*aj 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

in a command window. The output is,

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.