On transitivity of a product of groups

Question

I now that $S_n$ acts $n$-transitively on the set $X=\{1,\dots ,n\}$. What can I say about the group $S_n\times S_m$? Is it true or false that $S_n\times S_m$ acts $(n+m)$-transitively on a set of $n+m$ elements?

Answer 1

For $n>2$ and $m>2$ the answer is no (assuming $S_n\times S_m$ is acting faithfully) as the stabilizer of a point would have to have index $n+m$.

It's a nice (potentially a little tricky) exercise to show that any subgroup of $S_n\times S_m$ of index $n+m$ contains a normal subgroup of $S_n\times S_m$ so the action would not be faithful.

For $n=m=2$, there is actually a transitive faithful action of $S_n\times S_m$ on $\{1,2,3,4\}$ with image $\langle(1,2)(3,4),(1,3)(2,4)\rangle$, but this is not $4$-transitive.