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(True/False) Suppose :ZZ4, x→ 2x (mod 4). Then ker() = 42. (True/False) Suppose G is an abelian group. Then : G x G → G, (x, y) → xy-¹ is a group homomorphism. (True/False) Suppose G is a group. Then : G x G → G, (x, y) →xy is a group homomorphism.

(True/False) Suppose :ZZ4, x→ 2x (mod 4). Then ker() = 42. (True/False) Suppose G is an abelian group. Then : G x G → G, (x, y) → xy-¹ is a group homomorphism. (True/False) Suppose G is a group. Then : G x G → G, (x, y) →xy is a group homomorphism.

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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**Problem 1:** **Statement:** (True/False) Suppose \(\phi : \mathbb{Z} \to \mathbb{Z}_4\), defined by \(x \mapsto 2x \mod 4\). Determine if \(\ker(\phi) = 4\mathbb{Z}\). **Explanation:** This question asks whether the kernel of the homomorphism \(\phi\) is the set of all integers that are multiples of 4. --- **Problem 2:** **Statement:** (True/False) Suppose \(G\) is an abelian group. Consider the homomorphism \(\phi : G \times G \to G\) defined by \((x, y) \mapsto xy^{-1}\). Is \(\phi\) a group homomorphism? **Explanation:** This question investigates whether the mapping that sends pairs \((x, y)\) to the product of \(x\) and the inverse of \(y\) qualifies as a group homomorphism in an abelian group. --- **Problem 3:** **Statement:** (True/False) Suppose \(G\) is a group. Consider the homomorphism \(\phi : G \times G \to G\) defined by \((x, y) \mapsto xy\). Is \(\phi\) a group homomorphism? **Explanation:** This question examines whether the mapping that simply computes the product of \(x\) and \(y\) is a group homomorphism for any group \(G\). --- These problems are designed to test understanding of concepts in abstract algebra, particularly dealing with group homomorphisms and kernels.
Transcribed Image Text:**Problem 1:** **Statement:** (True/False) Suppose \(\phi : \mathbb{Z} \to \mathbb{Z}_4\), defined by \(x \mapsto 2x \mod 4\). Determine if \(\ker(\phi) = 4\mathbb{Z}\). **Explanation:** This question asks whether the kernel of the homomorphism \(\phi\) is the set of all integers that are multiples of 4. --- **Problem 2:** **Statement:** (True/False) Suppose \(G\) is an abelian group. Consider the homomorphism \(\phi : G \times G \to G\) defined by \((x, y) \mapsto xy^{-1}\). Is \(\phi\) a group homomorphism? **Explanation:** This question investigates whether the mapping that sends pairs \((x, y)\) to the product of \(x\) and the inverse of \(y\) qualifies as a group homomorphism in an abelian group. --- **Problem 3:** **Statement:** (True/False) Suppose \(G\) is a group. Consider the homomorphism \(\phi : G \times G \to G\) defined by \((x, y) \mapsto xy\). Is \(\phi\) a group homomorphism? **Explanation:** This question examines whether the mapping that simply computes the product of \(x\) and \(y\) is a group homomorphism for any group \(G\). --- These problems are designed to test understanding of concepts in abstract algebra, particularly dealing with group homomorphisms and kernels.
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