TITLE: What is the group generated by the conjugacy class containing $(12\ldots n)$ in $S_n$?
QUESTION [6 upvotes]: This question is motivated by this answer to a question about groups generated by conjugacy classes.
Let $n \geq 1$ and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$.  Define the n-cycle $\alpha=\pmatrix{1&2&\cdots &n}$, and let $\operatorname{Cl}(\alpha)=\{\beta \alpha \beta^{-1}:\beta \in S_n\}$ denote the conjugacy class of $\alpha$.

Question: What is the group $\langle \operatorname{Cl}(\alpha) \rangle$?

Observation 1: If $n$ is odd and $n \geq 3$, then $\alpha$ and its conjugates are even permutations (since they all have the same cycle structure), so $\langle \operatorname{Cl}(\alpha) \rangle$ contains only even permutations, and thus is a subgroup of the alternating group $A_n < S_n$.
Observation 2: Judging from some computations in GAP, it looks like $\langle \operatorname{Cl}(\alpha) \rangle=A_n$ for odd $n \geq 3$ and $\langle \operatorname{Cl}(\alpha) \rangle=S_n$ for even $n \geq 2$.

REPLY [3 votes]: If $S$ is a subset of a group $G$ satisfying
$\bullet$ $gSg^{-1}\subset S$ for all $g$ in $G$ and 
$\bullet$ $s^{-1}\in S$ for all $s$ in $S$, 
then the subgroup generated by $S$ is normal.