GROUP THEORY

Group Theory



Group Theory

Introduction

There are organisations that rely on groups. They expect employees to work in groups and expect that the company will perform well if they work in groups. Sometimes the behaviour of groups in a company is good towards other groups; they are competitive and strive to be the number one in the company. This helps the company in reaching its goals, but if the groups are not in harmony then it creates problems for the company. Teams in a company can be created if there are people who can motivate them and if they are able to handle teams. Such teams are beneficial for the company and help them to achieve their goals. There are instances where creation of a team is not possible. Such instances include employees who do not want to work with one another or people who do not have the nature to work in teams.

Group Theory

The study of symmetries possessed by a physical system. A general analysis of phenomena by their symmetries can effectively elucidate quite complicated behaviour. For example, before Kepler's discovery of elliptical planetary trajectories (1609), planets were thought to follow circular orbits because such trajectories were considered to be perfectly symmetric. Newton recognized that the fundamental symmetries of nature do not necessarily appear in the individual trajectories of particles, but rather in the family of all possible trajectories; that is, in the equations of motion themselves. One can conceptually apply symmetry operations to a star-planet system that leave it invariant. From these operations, much about the system can be deduced without knowing explicitly the solutions to the equations of motion. Newton's universal law of Gravitation, for instance, exhibits spherical symmetry.

The force of gravity due to the attraction of a planet to a star is the same for all positions that are equidistant from the star. Nevertheless, the possible trajectories of the planet include non-symmetric elliptical orbits with the star at one of the foci. Planets in these orbits speed up when approaching perihelion and slow down when approaching aphelion, which is consistent with a spherically symmetric force law. This behaviour was first formulated as one of Kepler's Laws of planetary motion and is a consequence of the conservation of angular momentum. Non-symmetrical elliptical orbits are general solutions to the equations of planetary motion.

The symmetrical symmetry of the system reveals itself indirectly as the conservation of angular momentum. This association of a dynamical symmetry with a conservation law was first suggested by A. E. Noether in 1918 and is called Noether's theorem (Stephen,2005, pp.232 - 242).

The symmetry operations on any physical system must possess the properties of a mathematical GROUP. Groups can be finite (like the group of rotations of an equilateral triangle) or infinite (for example the set of all integers, with addition used to combine the members). Groups may also be classed as continuous or discrete. An example of a continuous group is the group of all continuous translations of a point on a spherical ...