Let $G$ be a group and $g_0\in G$ be a fixed element of that group. I'm wondering if $g_0 \mapsto g_0^{-1}$ can always be extended to a group automorphism $G \to G$. My feeling tells me yes. For abelian groups the map $g \mapsto g^{-1}$ is already an automorphism, so there it's trivial. But what about non-abelian groups?
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You are asking whether for any element $g_0 \in G$, there exists an automorphism $\phi \in \operatorname{Aut}(G)$ such that $\phi(g_0) = g_0^{-1}$.
For abelian groups this is true, since we can choose $\phi$ to be the map $g \mapsto g^{-1}$. But for non-abelian groups the claim fails - see answers to this question (linked by Jeremy Rickard in the comments).
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