Action of Sn on by conjugation on the transpositions

Question

I know that the action of $S_n$ is n-transitive on $\{1,...,n\}$. Does this imply something about the action on the set of transpositions by conjugation? For example, is this action 2-transitive or primitive? I have shown that for $S_3$ it is primitive but for $S_4$ imprimitive. What can say for general n?

Answer 1

$\newcommand{\Span}[1]{\left\langle #1 \right\rangle}$For $n \ge 2$ the stabiliser of the transposition $(n-1, n)$ in $S_{n}$ is its centraliser, which is $$ \Span{ (n-1, n) } \times S_{n-2}. $$ This is maximal in $S_{n}$ for $n = 3$. For $n = 4$ this is $$ \Span{ (3 4) , (1 2) } < D < S_{4}, $$ for the Sylow $2$-subgroup $D = \Span{ (3 4) , (1 2) , (1 3) (2 4)}$ of $S_{n}$, so that the action is indeed not primitive.

Now it is known that for $n \ne 2 k$, the subgroup $S_{k} \times S_{n - k}$ of $S_{n}$ is maximal. Therefore the action is primitive for $n \ge 5$.

However, for $n \ge 4$ the action is not $2$-transitive, as once $g \in S_{n}$ centralises $(1 2)$, it cannot conjugate $(1 3)$ to $(3 4)$.