My question is related to this topic : Permutation of coordinates: how many linearly independent vectors will this generate?
I noticed that for $n \ge 3$, we have $A_nv = S_nv$ for all vectors $v$.
My question is : what are precisely the permutations groups $G$ for which $Gv = S_nv$ for all $v$ ?
We can show that transitivity is a necessary condition for such a group and, stronger, being primitive (see for example https://en.wikipedia.org/wiki/Primitive_permutation_group) is a necessary condition too.
By this work : https://www.researchgate.net/publication/222651063_Permutation_polytopes_and_indecomposable_elements_in_permutation_groups, it appears $2$-transitivity is a sufficient condition.
But do you see a necessary and sufficient condition ?
Thank you for your help ! :-)
