prove that a group of order 45 is abelian.
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prove that a group of order 45 is abelian. Suggestion. rewrite the group as a direct product of an abelian group and a cyclic group.
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- Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.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.26. Prove or disprove that if a group has an abelian quotient group , then must be abelian.
- True or false Label each of the following statements as either true or false, where is subgroup of a group. 4. The generator of a cyclic group is unique.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.Prove that if r and s are relatively prime positive integers, then any cyclic group of order rs is the direct sum of a cyclic group of order r and a cyclic group of order s.
- 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.True or false Label each of the following statements as either true or false, where is subgroup of a group. 6. If a subgroup of a group is abelian, then must be abelian.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.