Answered: 1. Let A be an abelian group, and let G… | bartleby

Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 31E: 31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the...

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I need help with number 1

1. Let A be an abelian group, and let G = A × A (Cartesian product), which also is a group. Let
6: GA, 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)
(c)
(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: (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.)

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