International Encyclopedia of Systems and Cybernetics

A set of elements provided with an internal composition rule with the three basic following conditions

- the rule is associative;

- it admits one neutral element e;

- any element of the set can be symmetrized: a*a

(adapted from L. CHAMBADAL, 1969, p.1 07).

The group operates transformations on the set of the general form: a*a =e. This implies that the set is closed and self-generating, i.e. that any operation on an element of the set gives another element of the set.

R.D. CARMICHAEL, gives some examples of groups:

"- the set of integers, positive and negative and zero, the rule of combination being ordinary addition, (the identity (i.e. neutral element) is zero; the inverse of an element is its negative);…

"- the set of all real numbers except zero, the rule of combination being ordinary multiplication (Here unity is the identical element, and the inverse of an element is its reciprocal)" (1956, p.17).

Another interesting example are the rotation groups of Euclidian geometry, as for example the group – i.e. the various types of rotations around a vertical axis – that conserves the symmetries of an equilateral triangle.

These transitions can be represented by a matrix.

More generally, the postulates of Euclidian geometry generate a group, as they correspond to a collection of rigid motions which may transform a geometric figure into itself.

CARMICHAEL also states: "The elements common to two finite groups G1 and G2 form a finite group H known as the common subgroup of G1 and G2." (p.46).

This is useful for the formalization of partial isomorphisms.

The group concept is akin to those of hypercycle and organizational closure, as observed by J. SINGH (1972).

Autopoiesis could thus possibly be modelized through group theory.

W.R. ASHBY enounced the following theorem: "The lines of behavior of a state-determined system define a group" (1960, p.244). Thus the concept of group is related to topological stability.

While mathematical groups are used to study the properties of symmetric systems, they can also be used to classify the ways symmetries can break.

"The late appearance of groups in science shows that a theory based on them could only have resulted from the modern mathematical method of generalization and abstraction, the method of thinking in terms of "system ".

"With such concepts as "set", "group", "ring", "field", mathematics reached a stage of great generality. The object of its study is no longer the special character of certain magnitudes, but the structure of whole domains. In this way it becomes possible to make statements that are valid for many different fields. For an over-all summary or synthesis of widely varied parts of mathematics, the notion of a group has become indispensable" (Adapted from H. BEHNKE & aI., 1987)