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# sol4 - Algebra Assignment 4 Solutions 1 In this problem...

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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 φ : G G given by φ ( g ) = g 5 is an automorphism. (Definition: An automorphism is an isomorphism from a group to itself.) Proof: φ is a homomorphism as φ ( xy ) = ( xy ) 5 = x 5 y 5 = φ ( x ) φ ( y ) for all x, y G φ is injective as the kernal K φ = { g G : φ ( g ) = e } = H = { e } by assumption, and we proved in class that φ

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sol4 - Algebra Assignment 4 Solutions 1 In this problem...

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