Doubly transitive group action

Symbol-free definition
A group action on a set is said to be doubly transitive or 2-transitive if the induced action on the set of ordered tuples of distinct elements, is transitive. In other words, given any two ordered tuples each having a pair of distinct elements, there is a group element taking one ordered tuple to the other.

Another way of saying this is that the stabilizers of any two points must intersect in a subgroup whose index in each is 1 less than the index of the subgroups.

Definition with symbols
A group action of a group $$G$$ on a set $$S$$ is said to be doubly transitive if given any $$(x,y)$$ and $$(x',y')$$ with $$x \ne y, x' \ne y'$$, all elements of $$S$$, there exists $$g \in G$$ such that $$g.x = x'$$ and $$g.y=y'$$.

Equivalently, it is doubly transitive if it is a transitive group action and if $$H$$ and $$K$$ are the stabilizers of two distinct points, and their index is $$n$$ (which is also the size of $$S$$, by transitivity) then $$H \cap K$$ has index $$n(n-1)$$.

Stronger properties

 * Weaker than::Triply transitive group action

Weaker properties

 * Stronger than::Doubly set-transitive group action
 * Stronger than::Primitive group action
 * Stronger than::Generously transitive group action
 * Stronger than::Transitive group action

Related group properties

 * Doubly transitive group is a group possessing a doubly transitive action