Construct a nontrivial homomorphism $: Zg → Zı5 >
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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![**Question (a):** Construct a nontrivial homomorphism
\[ \phi: \mathbb{Z}_6 \to \mathbb{Z}_{15} \]
**Question (c):** Find the \( \text{Ker}\phi \)
**Explanation:**
This question asks you to construct a nontrivial homomorphism from the cyclic group \(\mathbb{Z}_6\) to the cyclic group \(\mathbb{Z}_{15}\) and to find the kernel of this homomorphism. A nontrivial homomorphism is one that is not the zero homomorphism. The kernel of a homomorphism \(\phi\) is the set of elements in the domain that map to the identity element of the codomain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80fac62b-1f91-4b94-88b2-6d2c841a4645%2F81b8e996-ba9e-47dd-851c-9933b27e5115%2Frhyimxd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question (a):** Construct a nontrivial homomorphism
\[ \phi: \mathbb{Z}_6 \to \mathbb{Z}_{15} \]
**Question (c):** Find the \( \text{Ker}\phi \)
**Explanation:**
This question asks you to construct a nontrivial homomorphism from the cyclic group \(\mathbb{Z}_6\) to the cyclic group \(\mathbb{Z}_{15}\) and to find the kernel of this homomorphism. A nontrivial homomorphism is one that is not the zero homomorphism. The kernel of a homomorphism \(\phi\) is the set of elements in the domain that map to the identity element of the codomain.
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