When does a group presentation of a group $G = < x_{1} , \ldots , x_{n} | r_{i}(x_{1} , \ldots , x_{n} ), \; 1 \leq i \leq k> $ produce all the elements of $Aut(G)$ by permuting the $x_{i}$?
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$\begingroup$ Very rarely, I think it has to be either finite or be virtually Z. $\endgroup$– MishaCommented Apr 11, 2013 at 1:52
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$\begingroup$ Indeed, such group G would have to have finite automorphism group. Looking at the inner automorphisms, one immediately sees that G is either finite or is an extension of the infinite cyclic group with finite kernel: $1\to F \to G \to Z\to 1$, where F is finite. $\endgroup$– MishaCommented Apr 11, 2013 at 2:39
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$\begingroup$ I deleted my comment that the fundamental group of the Klein bottle is an example, I forgot that the Dehn twist makes the automorphism group infinite. $\endgroup$– Lee MosherCommented Apr 11, 2013 at 2:47
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