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Question: Z - Z is a group under componentwise addition and Z is a group under addition. ZXZ Prove that ≈Z. ((12,17)) Define f: ZXZ → Z by ƒ (x, y) = 17x - 12y.

Question: Z - Z is a group under componentwise addition and Z is a group under addition. ZXZ Prove that ≈Z. ((12,17)) Define f: ZXZ → Z by ƒ (x, y) = 17x - 12y.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Algebraic Structures 

11.2 The Isomorphism Theorems

**Question:** \(\mathbb{Z} \times \mathbb{Z}\) is a group under componentwise addition and \(\mathbb{Z}\) is a group under addition. Prove that \(\frac{\mathbb{Z} \times \mathbb{Z}}{\langle(12,17)\rangle} \approx \mathbb{Z}\). Define \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\) by \(f(x,y) = 17x - 12y\).
Transcribed Image Text:**Question:** \(\mathbb{Z} \times \mathbb{Z}\) is a group under componentwise addition and \(\mathbb{Z}\) is a group under addition. Prove that \(\frac{\mathbb{Z} \times \mathbb{Z}}{\langle(12,17)\rangle} \approx \mathbb{Z}\). Define \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\) by \(f(x,y) = 17x - 12y\).
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