Generously transitive group action

Symbol-free definition
A group action on a set is termed generously transitive if given any two elements in the set, there is an element of the group that acts to interchange the two elements.

Definition with symbols
A group action of a group $$G$$ on a set $$S$$ is termed generously transitive if given any two elements $$a$$ and $$b$$ in $$S$$, there is a $$g$$ in $$G$$ such that $$g.a = b$$ and $$g.b = a$$.

Stronger properties

 * Doubly transitive group action

Weaker properties

 * Transitive group action