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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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