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
Section4.6: Quotient Groups
Problem 27E:
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
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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.
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