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.

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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 9E: 9. Suppose that and are subgroups of the abelian group such that . Prove that .

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