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Questions about the branch of algebra that deals with groups.

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automorphism group and group presentation

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 $x_{i}$?