Much more is true, in that some transformation can take any given distinct points "Qi" for to any other 3-tuple "Ri" of distinct points ("triple transitivity"). This amount of specification 'uses up' the three dimensions of PGL2("K"); in other words, the group action is sharply 3-transitive. The computational aspect of this is the cross-ratio. Indeed, a generalized converse is true: a sharply 3-transitive group action is always (isomorphic to) a generalized form of a PGL2("K") action on a projective line, replacing "field" by "KT-field" (generalizing the inverse to a weaker kind of involution), and "PGL" by a corresponding generalization of projective linear maps.