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- Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.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.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
- Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.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 .If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.