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Unformatted text preview: Algebra , Assignment 4, Solutions 1. In this problem, assume (important!) that G is an Abelian . Set H = { g G : g 5 = e } . (Warning: Expressions such as x 1 / 5 are not well defined. Do not use them!) (a) Show H is a subgroup of G . Point out where the assumption that G was Abelian was used. Solution. As usual, three parts. Identity: As e 5 = e , e H . Product: If x,y H then x 5 = y 5 = e so that ( xy ) 5 = x 5 y 5 = ee = e so that xy H where we need that the group is Abelian to say that ( xy ) 5 = xyxyxyxyxy = xxxxxyyyyy = x 5 y 5 Inverse: If xinH then x 5 = e so that ( x 1 ) 5 = ( x 5 ) 1 = e 1 = e so x 1 H . (b) Show H is a normal subgroup of G . G is Abelian so that all subgroups of G are Normal! (c) Assume further that G is finite and that H = { e } . Show that the map : G G given by ( g ) = g 5 is an automorphism. (Definition: An automorphism is an isomorphism from a group to itself.) Proof: is a homomorphism as ( xy ) = ( xy ) 5 = x 5...
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This note was uploaded on 04/01/2011 for the course MATH 101 taught by Professor Josephs during the Fall '08 term at NYU.
 Fall '08
 JOSEPHS
 Algebra

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