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Problem 1. In each of the following problems, determine if the given is a group homomorphism. 1.1.4: Z/12Z → Z/6Z given by ([k]) = [2k] for all k Є Z. 1.2. Let G and H be any two groups, for all gЄ G and hЄ H. and let : Gx H → G be given by (g, h) = g 1.3. & : GL2(R) → GL2 (R), given by (A) = AT for all A = GL2(R). 1.4. Let G be any abelian group, and let : G → G be given by (g) = g2 for all g Є G. Remark. In Problem 1.1. you should explain whether or not is a function at all: is it well-defined? That is, if k and are both representatives of the same class in Z/12Z, do we have ([k]) = ([l])?

Problem 1. In each of the following problems, determine if the given is a group homomorphism. 1.1.4: Z/12Z → Z/6Z given by ([k]) = [2k] for all k Є Z. 1.2. Let G and H be any two groups, for all gЄ G and hЄ H. and let : Gx H → G be given by (g, h) = g 1.3. & : GL2(R) → GL2 (R), given by (A) = AT for all A = GL2(R). 1.4. Let G be any abelian group, and let : G → G be given by (g) = g2 for all g Є G. Remark. In Problem 1.1. you should explain whether or not is a function at all: is it well-defined? That is, if k and are both representatives of the same class in Z/12Z, do we have ([k]) = ([l])?

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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Transcribed Image Text:Problem 1. In each of the following problems, determine if the given is a group homomorphism. 1.1.4: Z/12Z → Z/6Z given by ([k]) = [2k] for all k Є Z. 1.2. Let G and H be any two groups, for all gЄ G and hЄ H. and let : Gx H → G be given by (g, h) = g 1.3. & : GL2(R) → GL2 (R), given by (A) = AT for all A = GL2(R). 1.4. Let G be any abelian group, and let : G → G be given by (g) = g2 for all g Є G. Remark. In Problem 1.1. you should explain whether or not is a function at all: is it well-defined? That is, if k and are both representatives of the same class in Z/12Z, do we have ([k]) = ([l])?

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