GROUP (in mathematical sense)
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 1471 ▶ |
| Object type | Methodology or model |
A Template:Ency term of Template:Ency term provided with an internal composition Template:Ency term with the three basic following conditions
- the Template:Ency term is associative;
- it admits one neutral Template:Ency term e;
- any Template:Ency term of the Template:Ency term can be symmetrized: a*a-1
(adapted from Template:Ency person, 1969, p.107).
The group operates Template:Ency term on the Template:Ency term of the general form: a*a-1 =e. This implies that the Template:Ency term is closed and self-generating, i.e. that any Template:Ency term on an Template:Ency term of the Template:Ency term gives another Template:Ency term of the Template:Ency term.
Template:Ency person, gives some examples of groups:
- “- the Template:Ency term of integers, positive and negative and zero, the Template:Ency term of combination being ordinary addition, (the identity (i.e. neutral element) is zero; the inverse of an element is its negative);…
- “- the Template:Ency term of all real numbers except zero, the Template:Ency term of combination being ordinary multiplication (Here unity is the identical Template:Ency term, and the inverse of an Template:Ency term is its reciprocal)” (1956, p.17).
Another interesting example are the Template:Ency term groups of Euclidian geometry, as for example the group — i.e. the various types of rotations around a vertical axis — that conserves the Template:Ency term of an equilateral triangle.
These Template:Ency term can be represented by a Template:Ency term.
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.
Template:Ency person also states: “The Template:Ency term 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 Template:Ency term.
The group Template:Ency term is akin to those of Template:Ency term and Template:Ency term, as observed by Template:Ency person (1972).
Template:Ency term could thus possibly be modelized through group theory.
Template:Ency person enounced the following theorem: “The lines of Template:Ency term of a Template:Ency term define a group” (1950, p.244). Thus the Template:Ency term of group is related to Template:Ency term.
While mathematical groups are used to study the properties of Template:Ency term systems, they can also be used to classify the ways Template:Ency term can break.
- “The late appearance of groups in science shows that a theory based on them could only have resulted from the modern mathematical Template:Ency term of Template:Ency term and Template:Ency term , the Template:Ency term of thinking in terms of ”system“.
- “With such concepts as ”Template:Ency term “, ”group“, ”ring“, ”Template:Ency term “, mathematics reached a stage of great generality. The object of its study is no longer the special character of certain Template:Ency term , but the Template:Ency term of whole Template:Ency term . In this way it becomes possible to make statements that are valid for many different Template:Ency term . For an over-all summary or synthesis of widely varied parts of mathematics, the notion of a group has become indispensable” (Adapted from Template:Ency person & al., 1987)