APP. OF GROUP ACTIONS

APP. OF GROUP ACTIONS - APPLICATIONS OF GROUP ACTIONS A...

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APPLICATIONS OF GROUP ACTIONS A group action is a way of describing symmetries in which every element of the group "acts" like a bijective transformation(homomorphism, linear, smooth, continuous) of some set (for example, a group, vector space, manifold, topological space), without being identified with that transformation. One can rotate a regular -sided polygon of a turn, or we can flip it over. One can rearrange the elements of a set. The common string in all these is a collection of reversible transformations. We often utilize the fact that important structures arise from families of morphisms that are indexed by a group. For example, rotations in the plane about the origin are indexed by the unimodular group of complex numbers; we say that this group acts on the plane and the orbit of a point at distance r from the origin is the circle of radius r . Historically, the first group action studied was the action of the Galois group on the roots of a polynomial . However, there are numerous examples and applications of group actions in many branches of science including algebra , topology , geometry , number theory , statistics and analysis . In this paper, I attempt to include at least one example of application of group action in
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This note was uploaded on 02/01/2010 for the course MATH 199 taught by Professor Staff during the Spring '08 term at E. Washington.

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APP. OF GROUP ACTIONS - APPLICATIONS OF GROUP ACTIONS A...

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