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We know that $\left\vert G\right\vert <\infty \Rightarrow \left\vert Aut\left( G\right) \right\vert <\infty $.

My question is: $\exp \left( G\right) <\infty \Rightarrow \exp \left( Aut\left( G\right) \right) <\infty ?$

Where $Aut\left( G\right) $=set automorphisms of $G$ and $\exp \left( G\right)$ is the exponent of G.

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    Next question: what if $G$ is finitely generated?2012-11-07

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No. Let $G$ be the product of infinitely many copies of $C_2$. Then $\text{Aut}(G)$ contains permutations of infinite order.

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    For example let the indexing set be $\mathbb{Z}$ and consider the permutation $n \mapsto n + 1$.2012-11-07