Problem 5. Let G be a group, and suppose that the quotient group G/Z(G) is cyclic. Show that G is abelian. Hint. Since G/Z(G) is cyclic, we can write G/Z(G) = [[g]) for some g = G. Show that every element of G must be of the form gaz for some a € Z and z = Z(G). Then show that any two elements of this form commute.

Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)

Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.

25. Prove or disprove that every group of order 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.

Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.

13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by

10. Suppose that and are subgroups of the abelian group such that . If is a subgroup of such that , prove that .

15. Prove that if for all in the group , then is abelian.

34. Suppose that and are subgroups of the group . Prove that is a subgroup of .