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Let $S_n$ be the symmetric group on $n$ elements and $\sigma \in S_n$ be a permutation. One can represent $\sigma$ (essentially uniquely) as a set of $m$ disjoint cycles of lengths $l_1,l_2,\dots,l_m$ such that these cycles partition the set of $n$ objects.

Does there exist a (non-trivial) group $G$ that acts (non-trivially) on $S_n$ that preserves the number of disjoint cycles and their respective lengths? I.e. does there exist a group, $G$, such that for any $g \in G$ if a permutation $\sigma \in S_n$ can be represented by a set of $m$ disjoint cycles of lengths $l_1,l_2,\dots,l_m$ then the permutation, $g \cdot \sigma$, can similarly be represented as a set of $m$ disjoint cycles of lengths $l_1,l_2,\dots,l_m$?

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Sure, $S_n$ itself, acting by conjugation. –  Tobias Kildetoft May 7 at 18:08

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