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For given integer $k>1$ and k-transposition $g \in S_n$, is there a isomorphism from symmetric group $S_n$ to itself such than sends $g$ to a transposition (a 2-cycle)?

It seems that for $k=2$ it is impossible: Is there an automorphism of symmetric group of degree 6 sending a transposition to product of two transpositions?

What about the same problem replacing "transposition" by "m-cycle"?!

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    $\begingroup$ This possible only when $n=6$ and $k=3$. For $n \ne 6$ it is not possible because ${\rm Aut}(S_n) \cong S_n$. For $n=6$, the outer automorphism exchanges $3$-cycles with products of $2$ $3$-cycles. $\endgroup$ – Derek Holt Mar 10 '16 at 9:01
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    $\begingroup$ @DerekHolt: That looks like an answer to me. $\endgroup$ – joriki Mar 10 '16 at 9:18
  • $\begingroup$ Because all automorphism of $S_n$ ($n \neq 6$) are inner, so every automorphism of $S_n$ preserves cycle type of permutations. Is it true? $\endgroup$ – user321571 Mar 10 '16 at 9:19
  • $\begingroup$ Yes that is correct - so you only need to consider the outer automorphism of $S_6$, which is unique modulo inner automorphisms. $\endgroup$ – Derek Holt Mar 10 '16 at 11:34

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