Question

Suppose that a, b ∈ Z+ are relatively prime. Show that ϕ : Z × Z → Z defined as ϕ((m, n)) = bm − an is a group homomorphism and Ker(ϕ) = ⟨(a, b)⟩. Conclude that (Z × Z)/⟨(a, b)⟩ ∼= Z.

Name: suppose that a b z are relatively prime show that z z z defined as m n bm an is a group homomorphism and ker a b conclude that z za b z 64598
Uploaded: 2024-08-24T19:01:46-08:00
Duration: 9 min 35 s
Channel: Sarah Gift
Description: suppose that a b z are relatively prime show that z z z defined as m n bm an is a group homomorphism and ker a b conclude that z za b z 64598

          Suppose that a, b ∈ Z+ are relatively prime. Show that ϕ : Z × Z → Z defined
as
ϕ((m, n)) = bm − an
is a group homomorphism and Ker(ϕ) = ⟨(a, b)⟩. Conclude that (Z × Z)/⟨(a, b)⟩ ∼= Z.

Added by Celia C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sarah Gift

08/24/2024

Step 1

** To prove that ϕ is a group homomorphism, we need to show that for any two elements (m1, n1) and (m2, n2) in Z × Z, the following holds: \[ ϕ((m1, n1) + (m2, n2)) = ϕ((m1 + m2, n1 + n2)) = b(m1 + m2) - a(n1 + n2). \] Calculating the right-hand side: \[ b(m1 + Show more…

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Suppose that a, b ∈ Z+ are relatively prime. Show that ϕ : Z × Z → Z defined as ϕ((m, n)) = bm − an is a group homomorphism and Ker(ϕ) = ⟨(a, b)⟩. Conclude that (Z × Z)/⟨(a, b)⟩ ∼= Z.