Chapter 5 Some Basic Techniques of Group Theory

Book Id: WPLBN0000673790
Format Type: PDF eBook:
File Size: 0.4 MB
Reproduction Date: 2005

Title:	Chapter 5 Some Basic Techniques of Group Theory
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Language:	English
Subject:	Science., Mathematics, Logic
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Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: Groups Acting on Sets. In this chapter we are going to analyze and classify groups, and, if possible, break down complicated groups into simpler components. To motivate the topic of this section, let?s look at the following result. Cayley?s Theorem Every group is isomorphic to a group of permutations. Proof. The idea is that each element g in the group G corresponds to a permutation of the set G itself. If x E G, then the permutation associated with g carries x into gx. If gx = gy, then premultiplying by g?1 gives x = y. Furthermore, given any h E G, we can solve gx = h for x. Thus the map x - gx is indeed a permutation of G. The map from g to its associated permutation is injective, because if gx = hx for all x E G, then (take x = 1) g = h. In fact the map is a homomorphism, since the permutation associated with hg is multiplication by hg, which is multiplication by g followed by multiplication by h, h ? g for short. Thus we have an embedding of G into the group of all permutations of the set G...