7. Find all distinct abelian groups of order 30. That is give a list of groups, and prove that all elements in your list are distinct.
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- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Exercises 35. Prove that any two groups of order are isomorphic.Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
- 9. Suppose that and are subgroups of the abelian group such that . Prove that .15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .9. Find all homomorphic images of the octic group.
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.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.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.