Problem 3. In this problem, we will write [b] for the element [b] € Z/kZ. 3.1. Show that the map : Z/mnZ (Z/mZ) × (Z/nZ), 4([a]mn) = ([a]m, [a]n) is well-defined, and is a group homomorphism. 3.2. Now suppose in addition that m and n are relatively prime. Determine if is an isomorphism. 3.3. If m and n are not relatively prime, can be an isomorphism? 4

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Problem 3. In this problem, we will write [b] for the element [b] € Z/kZ.
3.1. Show that the map
: Z/mnZ
(Z/mZ) × (Z/nZ), 4([a]mn) = ([a]m, [a]n)
is well-defined, and is a group homomorphism.
3.2. Now suppose in addition that m and n are relatively prime. Determine if
is an isomorphism.
3.3. If m and n are not relatively prime, can be an isomorphism?
4
Transcribed Image Text:Problem 3. In this problem, we will write [b] for the element [b] € Z/kZ. 3.1. Show that the map : Z/mnZ (Z/mZ) × (Z/nZ), 4([a]mn) = ([a]m, [a]n) is well-defined, and is a group homomorphism. 3.2. Now suppose in addition that m and n are relatively prime. Determine if is an isomorphism. 3.3. If m and n are not relatively prime, can be an isomorphism? 4
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