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Groups as symmetry

878 bytes added, 02:45, 1 March 2008
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Groups are everywhere. And it doesn't take a lot of effort to spot them, if you look. This article describes how one can locate groups easily. No knowledge except the definition of [[group]], [[subgroup]], [[trivial group]] and [[Abelian group]] is required.
==Groups as symmetriesSymmetry==
===Symmetry from a geometric perspective===
Symmetries of an object is measured by the set of transformations that map the object to itself. Object can be replaced by a structure, or rule. We're used to thinking of symmetries of concrete objects (like mice, clocks, and historical monuments). But in physics, we're interested in the symmetry and invariance properties enjoyed by ''laws''. In chemistry, we're interested in the symmetries of small things like molecules.
===Symmetry from a fairness perspective===
Let's go a little further with the idea of symmetry in laws, but this time, from the point of view of ''law'' in the sense of laws made by humans. We'd ideally like our laws to enjoy a certain kind of symmetry: a ''fairness''. For instance, if a set of people appear in a certain court case in a certain case, the outcome should be independent of race, gender, caste etc. of the people. So we'd like our law to be invariant under the ''symmetries'' of changing the ''people'' involved.
The same can be said about providing any service fairly: the level of service provided should be invariant under all the ways of ''permuting'' the people around.
This also relates to another important fact: equality, and fairness, translate in group theory to invariance under a certain group action. The group action is the one that permutes the elements.
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