Chapter5 - Chapter 5 Some Basic Techniques of Group Theory...

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Chapter 5 Some Basic Techniques of Group Theory 5.1 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. 5.1.1 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 G , then the permutation associated with g carries x into gx .I f = gy , then premultiplying by g 1 gives x = y . Furthermore, given any h G , we can solve = h for x . Thus the map x is indeed a permutation of G . The map from g to its associated permutation is injective, because if = hx for all x 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 . In Cayley’s theorem, a group acts on itself in the sense that each g yields a permutation of G . We can generalize to the notion of a group acting on an arbitrary set. 5.1.2 DeFnitions and Comments The group G acts on the set X if for each g G there is a mapping x of X into itself, such that 1
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2 CHAPTER 5. SOME BASIC TECHNIQUES OF GROUP THEORY (1) h ( gx )=( hg ) x for every g,h G (2) 1 x = x for every x X . As in (5.1.1), x deFnes a permutation of X . The main point is that the action of g is a permutation because it has an inverse, namely the action of g 1 . (Explicitly, the inverse of x is y g 1 y .) Again as in (5.1.1), the map from g to its associated permutation Φ( g ) is a homomorphism of G into the group S X of permutations of X . But we do not necessarily have an embedding. If = hx for all x , then in (5.1.1) we were able to set x = 1, the identity element of G , but this resource is not available in general. We have just seen that a group action induces a homomorphism from G to S X , and there is a converse assertion. If Φ is a homomorphism of G to S X , then there is a corresponding action, deFned by =Φ( g ) x,x X . Condition (1) holds because Φ is a homomorphism, and (2) holds because Φ(1) must be the identity of S X . The kernel of Φ is known as the kernel of the action ; it is the set of all g G such that = x for all x , in other words, the set of g ’s that Fx everything in X . 5.1.3 Examples 1. ( The regular action ) Every group acts on itself by multiplication on the left, as in (5.1.1). In this case, the homomorphism Φ is injective, and we say that the action is faithful . [Similarly, we can deFne an action on the right by ( xg ) h = x ( gh ), x 1= x , and then G acts on itself by right multiplication. The problem is that Φ( )=Φ ( h ) Φ( g ), an antihomomorphism. The damage can be repaired by writing function values as xf rather than f ( x ), or by deFning the action of g to be multiplication on the right by g 1 .W e will avoid the difficulty by restricting to actions on the left.] 2. (
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This note was uploaded on 11/14/2011 for the course MATH 367 taught by Professor Sdd during the Spring '11 term at Middle East Technical University.

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Chapter5 - Chapter 5 Some Basic Techniques of Group Theory...

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