Definition:Transitive Group Action/n-transitive

Definition
Let $G$ be a group.

Let $S$ be a set.

Let $*: G \times S \to S$ be a group action. Let $n\geq1$ be a natural number.

The group action is $n$-transitive for any two ordered $n$-tuples $(x_1, \ldots, x_n)$ and $(y_1, \ldots, y_n)$ of pairwise distinct elements of $S$, there exists $g\in G$ such that:
 * $\forall i\in \{1, \ldots, n\} : g * x_i = y_i$