Chapter 7: 57E (page 213)
Show that the group U18 of the units in Z18is cyclic.
It has been proved that U18is cyclic
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Question: Let be an isomorphism of groups. Let be the inverse function off as defined in Appendix B. Prove that g is also an isomorphism of groups. [Hint: To show that , consider the images of the leftand right-hand sides under f and use the facts that f is a homomorphism and is the identity map.]
Question: Let N be a subgroup of a group G and let .
(b) Prove that is N isomorphic to . [Hint: Define by ]
Question: Prove that
Let G and H be groups. If is a cyclic group, prove that G and H are both cyclic. (Exercise 12 shows that the converse is false)
Show that the additive group is cyclic.
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