A group G of homeomorphisms
of a topological space X onto itself is called n-transitive if any set of n points in X
can be mapped onto any other set of n points by some member of G. In this paper,
we investigate the transitivity of G when X is euclidean m-space Em or real
projective m-space Πm, and G properly contains the group Am of affine
transformations or the group Pm of projective transformations, respectively. We show
that G ⊃ A1 implies that G is at least 3-transitive, G ⊃ P1 implies that G is at least
4-transitive, and, for a fairly wide class of groups, G is n-transitive for every n. For
higher dimensional spaces, our information is considerably more meager. We show
that G ⊃ Am or G ⊃ Pm implies that G is at least 3-transitive, and that if some
member of G leaves fixed the points of some open set, then G is n-transitive for every
n.
