Homomorphisms
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Homomorphism
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Consider S3 the group of symmetries of the equilateral triangle. We have seen that it has the following combination table. [Diagram goes here - download the original to see it.] The partition shows that the group can be divided into four quadrants as follows: [Diagram goes here - download the original to see it.] where R stands for any rotation and Q stands for any reflection. The set {R, Q} is a group under the operation of composition of transformations. This structure is isomorphic to ¢2 the cyclic group of two elements. Thus ¢2 is a substructure of S3 . In order to capture this concept of one substructure manifested in another we define a homomorphism. [Equation goes here - download the original to see it.] Thus, the homomorphism property is the same property that preserves structure for isomorphisms. The difference is that isomorphisms are special cases of homomorphisms - they are homomorphisms such that the order of G is equal to the order of H | G| = |H| This means that there are exactly the same number of elements in G as there are in H. That is, isomorphisms are one-one correspondences (bijections) but homomorphisms are many-one correspondences. Isomorphisms are specials cases of homomorphisms. An isomorphism is a bijective homomorphism. For S3 the homomorphism from ¢2 to ¢2 maps rotations of S3 to 0 and reflections of S3 to 1. [Diagram goes here - download the original to see it.] The homomorphism property can be verified for any pair of elements of S3
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Contents of
Homomorphisms
1 Homomorphism
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