Sharp t-transitivity ($|G|=n(n-1)\cdots (n-t+1)$)

An action is sharply $t$-transitive if each ordered $t$-tuple of distinct points is carried to any other by exactly one group element. For finite degree $n\ge t$, its order is $n(n-1)\cdots (n-t+1)$ and each ordered-tuple stabilizer is trivial. Sharp one-transitivity is a regular action. Finite sharp two-transitivity gives a regular kernel of a finite sharply two-transitive group.

Exactly one element sends each ordered triple of distinct points to each other such triple. On a projective line, representatives of the first two target points form a basis; scaling them so their sum represents the third point gives the unique projective transformation.

A group action is sharply two-transitive when exactly one element sends each ordered distinct pair to each other ordered distinct pair. The affine group over a finite field provides an example: solve $ax+b=u$ and $ay+b=v$ uniquely, giving $a=(v-u)/(y-x)$.

In a finite sharply two-transitive group, the identity together with all fixed-point-free elements forms a regular normal subgroup. One proof constructs virtual characters from induced point-stabilizer characters and the augmentation character, proves their norms are one, and realizes this set as an intersection of representation kernels. Conjugation by the point stabilizer is transitive on the kernel's nonidentity elements. This forces the kernel to be elementary abelian, so it is a unique Sylow subgroup and is characteristic. Closure of the fixed-point-free set is a theorem, not something supplied merely by counting its size.