Vol. 23, No. 1, 1967

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Multiply transitive groups of transformations

James Victor Whittaker

Vol. 23 (1967), No. 1, 189–207

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.

Mathematical Subject Classification
Primary: 54.80
Received: 11 April 1966
Published: 1 October 1967
James Victor Whittaker