Che Prove: If Aut(G) is cyclic, then so is any subgroup of it, in particular Inn(G). Inn(G) G/Z(G) where Z(G) is the center. If G/Z(G) iscyclic, the group is abelian.

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Answered: Prove: If Aut(G) is cyclic, then so is…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 30E: Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.

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Prove: If Aut(G) is cyclic, then so is any subgroup of it, in particular Inn(G). Inn(G) G/Z(G) where Z(G) is the center. If G/Z(G) is cyclic, the group is abelian.