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6. Use the Fundamental Homomorphism Theorem to prove that the following groups are isomorphic. (a) Z3 Z3 × Z3/K where K = = {(0, 0), (1, 1), (2, 2)}. Hint: Consider the function f(a, b) = a - b from Z3 × Z3 to Z3 and show that it is an onto-homomorphism. = (b) For any abelian group G, prove that H = {x² : x € G} ≈ G/K where K = {x € G : x² = e}. Hint: Consider the function f: G→ H defined by f(x) = x².

6. Use the Fundamental Homomorphism Theorem to prove that the following groups are isomorphic. (a) Z3 Z3 × Z3/K where K = = {(0, 0), (1, 1), (2, 2)}. Hint: Consider the function f(a, b) = a - b from Z3 × Z3 to Z3 and show that it is an onto-homomorphism. = (b) For any abelian group G, prove that H = {x² : x € G} ≈ G/K where K = {x € G : x² = e}. Hint: Consider the function f: G→ H defined by f(x) = x².

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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6. Use the Fundamental Homomorphism Theorem to prove that the following groups are isomorphic. (a) Z3 ≈ Z3 × Z3/K where K = {(0, 0), (1, 1), (2, 2)}. Hint: Consider the function f(a, b) = a · 6 from Z3 × Z3 to Z3 and show that it is an onto-homomorphism. (b) For any abelian group G, prove that H = {x² : x ¤ G} ≈ G/K where K = {x € G : x² = e}. Hint: Consider the function ƒ : G → H defined by f(x) = x².
Transcribed Image Text:6. Use the Fundamental Homomorphism Theorem to prove that the following groups are isomorphic. (a) Z3 ≈ Z3 × Z3/K where K = {(0, 0), (1, 1), (2, 2)}. Hint: Consider the function f(a, b) = a · 6 from Z3 × Z3 to Z3 and show that it is an onto-homomorphism. (b) For any abelian group G, prove that H = {x² : x ¤ G} ≈ G/K where K = {x € G : x² = e}. Hint: Consider the function ƒ : G → H defined by f(x) = x².
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