aa_hw_4_Math 113 - Homework 4 October 4 2010 Introduction...

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Introduction to Abstract Algebra Homework 4 October 4, 2010 Problem 1 Suppose the group G acts on the set X and x X . (a) If y is in the kernel, then x X : y · x = x . By definition of a group action, the identity 1 is in the kernel. Assume y is in the kernel, then x X : x = 1 · x = ( y - 1 y ) · x = y - 1 · ( y · x ) = y - 1 · x, so y - 1 is also in the kernel, meaning we have closure under inverses. Now assume that y and z are elements in the kernel, then ( yz ) · x = y · ( z · x ) = y · x = x, so ( xy ) is also in the kernel, giving us closure. Thus, the kernel of the action is a subgroup. (b) First, the identity 1 is in G x , because 1 · x = x by the definition of an action. Assume y G x , then 1 · x = ( y - 1 y ) · x = y - 1 · ( y · x ) = y - 1 · x, so also y - 1 G x , meaning we have closure under inverses. Assume y,z G x , then ( yz ) · x = y · ( z · x ) = y · x = x, so also xy G x meaning we have closure, thus G x is a subgroup. Problem 2 Clearly, since H is a subgroup in itself, all we really want to show in both cases is that H is a subset of N G ( H ) and C G ( H ) respectively. (a) To show that H is a subgroup of N G ( H ), we want to show that all elements of H are also in N G ( H ). Pick some h H , and consider two arbitrary h 1 ,h 2 H , we shall see that hHh - 1 = H , because hh 1 h - 1 = hh 2 h - 1 h 1 = h 2 : hh 1 h - 1 = hh 2 h - 1 h - 1 hh 1 h - 1 = h - 1 hh 2 h - 1 h 1 h - 1 = h 2 h - 1 h 1 h - 1 h = h 2 h - 1 h h 1 = h 2 . 1/4
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Introduction to Abstract Algebra Homework 4 October 4, 2010 Thus, N G ( H ) contains at least all of H , and H N G ( H ). In the case where H was not a subgroup of G , we might have 1 Ó∈ H , in which case H cannot be a suggroup of N G ( H ). (b) We wish to show that H C G ( H ) H is abelian. Clearly, if G is abelian so is H . Again, since H is already a subgroup, we just need to show H C G ( H ) H is abelian. Again, we check each
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This note was uploaded on 10/15/2011 for the course CHEM 1 taught by Professor Gelfand during the Summer '11 term at Solano Community College.

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aa_hw_4_Math 113 - Homework 4 October 4 2010 Introduction...

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