3. For any a e Z, let [ale denote [a] in Ze and let [al2 denote [a] in Z2. Given that the mapping0: Z6 Z2 defined by 0((a]e) = [a]2 is a homomorphism.(a) Find ker 0.(b) Find Zg/kere.(c) Find an isomorphism from Z6/kere to Z2.

Answered: Image /qna-images/answer/1164a7cc-c90b-4197-95f6-3e5baa4d4b45.jpg

3. For any a eE Z, let als denote (a] in Z6 and let (a)2 denote (a] in Z2. Given that the mapping 0: Z6 Z2 defined by 0(lale) = [a]2 is a homomorphism. %3D (a) Find ker 0. (b) Find Zg/ker0. (c) Find an isomorphism from Z6/kere to Z2.

3. For any a eE Z, let als denote (a] in Z6 and let (a)2 denote (a] in Z2. Given that the mapping 0: Z6 Z2 defined by 0(lale) = [a]2 is a homomorphism. %3D (a) Find ker 0. (b) Find Zg/ker0. (c) Find an isomorphism from Z6/kere to Z2.

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

3. For any a e Z, let [ale denote [a] in Ze and let [al2 denote [a] in Z2. Given that the mapping
0: Z6 Z2 defined by 0((a]e) = [a]2 is a homomorphism.
(a) Find ker 0.
(b) Find Zg/kere.
(c) Find an isomorphism from Z6/kere to Z2.

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