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Let $X$ be a finite set. Let $S$ be the symmetric group of $X$. Suppose $S$ acts on itself by inner automorphisms. Is it true that any two transpositions of $X$ are conjugates under this action?

user754614
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    Just a note to say that the inner automorphisms of the symmetric groups correspond to a relabelling of the underlying elements which are being permuted, and leaves the cycle types unchanged. [relabelling according to the permutation used for conjugation, or its inverse, depending on the conventions being adopted] Of the finite symmetric groups only $S_6$ has additional outer automorphisms. – Mark Bennet Feb 26 '20 at 18:58

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Welcome to MSE ^_^

yes - recall two permutations are conjugate iff they have the same cycle structure. Since the transpositions are exactly the 2-cycles, the claim follows.

Here is some discussion regarding the theorem I'm citing, in case you haven't seen it before.

I hope this helps ^_^

Chris Grossack
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