If G is Abelian group

Question

Problem in book is, If G is a group with center Z(G), and if G/Z(G) is cyclic, then G must be abelian. I konw how prove this but my question if G is abelian then Z(G)= G then there is no meaning of problem, am I write if not please explain me.

The trivial group is cyclic. – Randall Oct 25 '18 at 16:48 — Randall, Oct 25 '18 at 16:48

RhythmInk · Accepted Answer · 2018-10-25T17:20:48.143

1

You are misunderstanding the phrasing of the question. In the case where $Z(G)=G$ the proposition you stated is true $G/G$ is the trivial group which is indeed cyclic. The statement is IF $G/Z(G)$ is cyclic, then $G$ must be abelian. Not If $G$ is abelian AND $G/Z(G)$ is cyclic, then $G$ is abelian. Do you see the difference?

edited Oct 25 '18 at 17:20

answered Oct 25 '18 at 17:00

RhythmInk

3,140

No, I didn't see the difference. In proof we prove that G is abelian so in statement always G/Z(G) is trivial group. – user499117 Oct 26 '18 at 02:31
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We aren't assuming $G$ is abelian. We are showing $G$ is abelian. – RhythmInk Oct 26 '18 at 02:35
Ok thank you for help. – user499117 Oct 26 '18 at 02:51

If G is Abelian group

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