What does it mean for a group action to be sharply transitive?

Question

I have just become comfortable with the concept of K-transitivity of group actions. Now, as I begin to read more, I find some literature referencing a new term, "Sharply N-Transitive," where "N" is usually explicitly put. Below is a link containing such language.

My question: What is the difference between an action being sharply transitive and just plain old transitivity? Why do we care? Please try and keep explanations to the fundamentals, I am new to Group Theory.

https://en.wikipedia.org/wiki/Mathieu_group#Constructions_of_the_Mathieu_groups

Answer 1

You can find this on the Wikipedia page for group actions. $G$ is sharply $n$-transitive if for any two sequences of $n$ distinct points $ x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ there is exactly one $g\in G$ with $x_i^g=y_i$ (or $g\cdot x_i=y_i$ depending on your convention) for each $i$.