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(1) Define the product group of (G, o) and (H, x) as the set G x H = {(g, h): gG, he H} with composition where (91, h1) (92, h₂) = (910 92, h₁ x h₂). Prove that G x H this is a group, and that it is isomorphic to H x G. What is the identity element? What is the inverse of (g, h)?

(1) Define the product group of (G, o) and (H, x) as the set G x H = {(g, h): gG, he H} with composition where (91, h1) (92, h₂) = (910 92, h₁ x h₂). Prove that G x H this is a group, and that it is isomorphic to H x G. What is the identity element? What is the inverse of (g, h)?

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) Define the product group of (G, o) and (H, x) as the set Gx H = {(g, h): g = G, he H} with composition where (91, h1) (92, h₂) = (910 92, h₁ x h₂). Prove that G x H this is group, and that it is isomorphic to H x G. What is the identity element? What is the inverse of (g, h)?
Transcribed Image Text:(1) Define the product group of (G, o) and (H, x) as the set Gx H = {(g, h): g = G, he H} with composition where (91, h1) (92, h₂) = (910 92, h₁ x h₂). Prove that G x H this is group, and that it is isomorphic to H x G. What is the identity element? What is the inverse of (g, h)?
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