Let G(a) be a cyclic group of order n. For each integer m, define a mapfm : G -> G by fm(x)т= x"" for every x E G. Prove that(1) fm is a group homomorphism.(2) fm is an automorphism if and only if gcd(m, n) = 1.(3) Find the kernel and image of f4 whenn =тт10.

Let G=<a> be a cyclic group of order n. for any integer m defines:

Answered: Let G (a) be a cyclic group of order n.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 30E: Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G...

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Let G (a) be a cyclic group of order n. For each integer m, define a map fm : G -> G by fm(x) т = x"" for every x E G. Prove that (1) fm is a group homomorphism. (2) fm is an automorphism if and only if gcd(m, n) = 1. (3) Find the kernel and image of f4 whenn = т т 10.