# Are all the endomorphisms of a group generated by a conjugation?

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?

No, take $G= ((0,\infty ) ,\cdot )$ and $f(t) = t^2 .$
• ... or any non-trivial autodomorphism of an abelian group, e.g. $x\mapsto -x$ – Hagen von Eitzen Dec 12 '15 at 14:21