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I am trying to understand the proof of the following question:

If automorphisms of $G$ are cyclic, then $G$ is abelian.

I found the following answer here If $G$ is non-Abelian, show that $Aut(G)$ is not cyclic. :

Suppose that ${\rm Aut}\,(G)$ is cyclic and let $f$ its generator, let $x,y\in G$ such that $y$ does not commute with $yxy^{-1}$, (since $G$ is not commutative, you can find $z$ such that $z$ does not commute with $y$, write $x=y^{-1}zy, yxy^{-1}=z$) there exists $n$ such that $c_x(z)=xzx^{-1}=f^n, m: c_y(z)=yzy^{-1}=f^m$, this implies that $c_x$ commutes with $c_y$, we deduce that $c_y(c_x(x))=c_x(x))$, i.e $yxy^{-1}=xyxy^{-1}x^{-1}$, i.e $(yxy^{-1})x=x(yxy^{-1})$ i.e $x$ commutes with $yxy^{-1}$. Contradiction.

But I really do not understand this solution, could someone clarify it for me please?

EDIT: I am not searching for a proof only, I am searching for an easy proof that do not require previous theorems and I think the above proof do this, so the suggested duplicate does not answer my question, I hope the question is reopened for answers please.

Shaun
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Brain
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    This proof doesn't even use the fact that Aut(G) is cyclic. He defines two conjugation automorphisms $c_x$ and $c_y$, they commute so sub $y$ into $c_x\circ c_y=c_y\circ c_x$ to get $xyx^{-1}=yxyx^{-1}y^{-1}$, i.e. $y(xyx^{-1})=(xyx^{-1})y$, true for every $x,y$. Now for any $y,z\in G$, there exist $x\in G$ such that $xyx^{-1}=z$, hence it's commutative. – Tychus Findlay Nov 18 '21 at 05:00
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    The method you posted is not right at all. It deduces by contradiction that $x$ commutes with $yxy^{-1}$, but the original assumption it states is about $y$ instead of $x$ – Display name Nov 18 '21 at 05:47
  • @copingroidcel are you saying that it is not correct proof? – Brain Nov 18 '21 at 10:20
  • No, I was clarifying it like you asked, just rephrased his proof in clearer way. Actually I think there's an error, the last sentence is false, in general conjugation is not transitive. No wonder the cyclicality of Aut(G) is not used. – Tychus Findlay Nov 18 '21 at 11:14

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