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])?
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 5E
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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])?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2Feb6c6cc1-bf27-4f09-a8d1-86914518ac46%2Fxqxoi3er_processed.png&w=3840&q=75)
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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