Let G be a cyclic group with generator a, and let G' be a group isomorphic to G. If</> : G -+ G' is an isomorphism, show that, for every x E G, (x) is completely determined by the value </>(a). That is, if</> : G -+ G' and 1fr : G-+ G' are two isomophisms such that </>(a)= lfr(a), then </>(x) = 1/r(x) for all x E G.

Let G be a cyclic group with generator a, and let G' be a group isomorphic to G. If</> : G -+ G' is an isomorphism, show that, for every x E G, (x) is completely determined by the value </>(a). That is, if</> : G -+ G' and 1fr : G-+ G' are two isomophisms such that </>(a)= lfr(a), then </>(x) = 1/r(x) for all x E G.

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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