2. Show an Abelian group order 15 is cyclic.
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- Find two groups of order 6 that are not isomorphic.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.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.
- Prove that any group with prime order is cyclic.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.Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.
- 9. Find all homomorphic images of the octic group.Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.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 .
- 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.If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.9. Suppose that and are subgroups of the abelian group such that . Prove that .