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