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It is well known that for a group $G$, the mapping from $G$ to $G $

$$ r \rightarrow s r s^{-1} $$

generated by an arbitrary element $s \in G $ is an endomorphism of $G$.

The question is then, are all the endomorphisms of $G$ generated in this way?

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No, take $G= ((0,\infty ) ,\cdot )$ and $f(t) = t^2 .$

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    $\begingroup$ ... or any non-trivial autodomorphism of an abelian group, e.g. $x\mapsto -x$ $\endgroup$ – Hagen von Eitzen Dec 12 '15 at 14:21

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