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It is shown that every point transformation group whose prolonged orbit dimensions pseudo-stabilize at order 0 is equivalent, under a change of variables, to the elementary similarity group consisting of translations and dilatations. In the study of dierential invariants and symmetry groups of dierential equations, the geometry of the prolonged group actions plays an essential role. The purpose of this paper is to further elucidate one aspect of this geometry, namely to provide a complete characterization of groups which "pseudo-stabilize" at the lowest order. This result will help explain an observation in (5), which was based on Lie's classification of transformation groups in the plane, (1), (3), that, in the two-dimensional case, there is essentially only one group which has this anomalous behavior. In this paper, this observation will be extended to groups in an arbitrary number of independent and dependent variables. The reader should be familiar with the fundamentals of the Lie theory of prolongation of transformation groups on jet bundles, as presented, for instance, in my books (3), (5). I shall employ the same basic notation here. Let G be a connected r-dimensional local transformation group acting on an open subset M X ◊ U ' R p ◊R q of the space (bundle) coordinatized by p independent and q dependent variables. The space of infinitesimal generators of G — its Lie algebra — will be denoted by g. The action of G on the sections u = f(x) of M induces an action of G on the associated n-th order jet bundle J n = J n M, known as the n-th prolonged transformation group, and denoted by G (n) . The generic or maximal dimension of the orbits of G (n) is known as the prolonged orbit dimension (of order n) and denoted by sn. By definition, G (n) acts semi-regularly (meaning all orbits have the same dimension) on the open subset V n J n consisting of all points contained in orbits of maximal dimension.

Pseudo-Stabilization of Prolonged Group Actions I. The Order Zero Case