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- 27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.Describe all subgroups of the group under addition.
- 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.Suppose that G is a finite group. Prove that each element of G appears in the multiplication table for G exactly once in each row and exactly once in each column.Exercises 12. Prove that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. (Hint: Consider the mapping defined by for all .)
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.3. Consider the group under addition. List all the elements of the subgroup, and state its order.Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.