Let $G$ be a finite group of order $n$ and let $a in G$ be an element of order $k$. Suppose that $k$ and $n$ are relatively prime. Prove that the map $varphi: G ightarrow G$ defined by $varphi(g) = a^{ell} g a^{-ell}$ for some fixed $ell in mathbb{Z}$ is an automorphism of $G$.

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