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?
2026-03-29 08:45:10.1774773910
Are all transpositions of a finite set conjugate to each other under action by inner automorphisms?
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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 ^_^