if it acts transitively on the set of distinct ordered pairs
That is, assuming (without a real loss of generality) that
The group action is sharply 2-transitive if such
Similarly we can define sharply 2-transitive group.
, since the induced action on the distinct set of pairs is
The definition works in general with k replacing 2.
Such multiply transitive permutation groups can be defined for any natural number k. Specifically, a permutation group G acting on n points is k-transitive if, given two sets of points a1, ... ak and b1, ... bk with the property that all the ai are distinct and all the bi are distinct, there is a group element g in G which maps ai to bi for each i between 1 and k. The Mathieu groups are important examples.
Every group is trivially 1-transitive, by its action on itself by left-multiplication.
be the symmetric group acting on
, then the action is sharply n-transitive.
The group of n-dimensional similarities acts 2-transitively on
The group of n-dimensional projective transforms almost acts sharply (n+2)-transitively on the n-dimensional real projective space
The almost is because the (n+2) points must be in general linear position.
In other words, the n-dimensional projective transforms act transitively on the space of projective frames of
Every Zassenhaus group is 2-transitive, but not conversely.
The solvable 2-transitive groups were classified by Bertram Huppert and are described in the list of transitive finite linear groups.