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With inspirations from the proof of Barrington's Theorem, I have the following questions about symmetric groups $S_n \; (n \ge 2)$.

  1. Enumerate all proper subsets $T$ of $S_n$ satisfying that $\; \forall g,h \in T, \; \exists \phi \in S_n, \; h = \phi g \phi^{-1}$.

We know that $\{ e \}$ and the set of all $n$-cycles are satisfying subsets.

Edit #1: I've realized that those satisfying subsets are subsets of conjugacy classes of $S_n$. Particularly, a conjugacy class of $S_n$ is a set of all $n-$permutations decomposing into cycles with length $a_1,...,a_k$ given beforehand. So the first question is solved, the second question still stands.

Edit #2: For convenience, in question 2, we only consider the conjugacy classes.

  1. Among those satisfying subsets, enumerate all subsets $U$ of $S_n$ satisfying that $\; \exists g,h \in U, \; e \neq ghg^{-1}h^{-1} \in U$, whereas $e$ is the identity of $S_n$.

We know that the set of all $m$-cycles is a satisfying subset with $m$ odd $\ge 3, \; n \ge 5$. Also, this property is perhaps related to the commutator subgroup, a.k.a. the alternating group $A_n \; (n \ge 5)$.

In fact, we know that those satisfying subsets only exists when $n \ge 5$. Particularly when $n = 5$, the only satisfying subsets are the set of all $5-$cycles and the set of all $3-$cycles.

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    $\begingroup$ For the first question, the answer isn't all conjugacy classes but rather all subsets of conjugacy classes. $\endgroup$ – joriki Aug 3 '18 at 15:23
  • $\begingroup$ @joriki Yes, you're right. However, for convenience, perhaps only consider the classes in the second question. $\endgroup$ – Vincent J. Ruan Aug 3 '18 at 15:24

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