(14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd = e for some divisor d of n.
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- If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.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.
- 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 .25. Prove or disprove that every group of order is abelian.Let H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.
- 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. Find all homomorphic images of the octic group.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .