Multiply transitive group action

Every group is trivially sharply 1-transitive, by its action on itself by left-multiplication. Let S_n be the symmetric group acting on \{1, ..., n\}, then the action is sharply n-transitive. The group of n-dimensional similarities acts 2-transitively on \mathbb{R}^n. In the case n=1 this action is sharply 2-transitive, but for n>1 it is not. The group of n-dimensional projective transforms almost acts sharply (n+2)-transitively on the n-dimensional real projective space \mathbb{RP}^n. 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 \mathbb{RP}^n. ==Classifications of 2-transitive groups ==

Every 2-transitive group is a primitive group, but not conversely. 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. The insoluble groups were classified by using the classification of finite simple groups and are all almost simple groups. == See also ==