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1. Let A be an abelian group, and let G = A x A (Cartesian product), which also is a group. Let 6: G→ A, o(a, b) = ab (product in A). Prove that is surjective. Prove that is a homomorphism. Find the kernel of o (and verify your answer). Where did you use that A is abelian? (a) (b) (d) 2. Let (G, *) and (G', $) be groups, and let : G→ G' be a homomorphism. Let K be a subgroup of G'. Let H = {ge G: 0(g) = K}. Prove that H is a subgroup of G. (Pay close attention to details. Please use and $ for the binary operations, not just multiplication.)

1. Let A be an abelian group, and let G = A x A (Cartesian product), which also is a group. Let 6: G→ A, o(a, b) = ab (product in A). Prove that is surjective. Prove that is a homomorphism. Find the kernel of o (and verify your answer). Where did you use that A is abelian? (a) (b) (d) 2. Let (G, *) and (G', $) be groups, and let : G→ G' be a homomorphism. Let K be a subgroup of G'. Let H = {ge G: 0(g) = K}. Prove that H is a subgroup of G. (Pay close attention to details. Please use and $ for the binary operations, not just multiplication.)

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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1. Let \( A \) be an abelian group, and let \( G = A \times A \) (Cartesian product), which also is a group. Let \( \phi : G \to A \), \( \phi(a, b) = ab \) (product in \( A \)). - (a) Prove that \( \phi \) is surjective. - (b) Prove that \( \phi \) is a homomorphism. - (c) Find the kernel of \( \phi \) (and verify your answer). - (d) Where did you use that \( A \) is abelian? 2. Let \( (G, \ast) \) and \( (G', \$) \) be groups, and let \( \phi : G \to G' \) be a homomorphism. Let \( K \) be a subgroup of \( G' \). Let \( H = \{ g \in G : \phi(g) \in K \} \). Prove that \( H \) is a subgroup of \( G \). (Pay close attention to details. Please use \( \ast \) and \( \$ \) for the binary operations, not just multiplication.)
Transcribed Image Text:1. Let \( A \) be an abelian group, and let \( G = A \times A \) (Cartesian product), which also is a group. Let \( \phi : G \to A \), \( \phi(a, b) = ab \) (product in \( A \)). - (a) Prove that \( \phi \) is surjective. - (b) Prove that \( \phi \) is a homomorphism. - (c) Find the kernel of \( \phi \) (and verify your answer). - (d) Where did you use that \( A \) is abelian? 2. Let \( (G, \ast) \) and \( (G', \$) \) be groups, and let \( \phi : G \to G' \) be a homomorphism. Let \( K \) be a subgroup of \( G' \). Let \( H = \{ g \in G : \phi(g) \in K \} \). Prove that \( H \) is a subgroup of \( G \). (Pay close attention to details. Please use \( \ast \) and \( \$ \) for the binary operations, not just multiplication.)
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