Logical asides on the maximality of a subgroup

Abstract

We point out the geometric significance of a part of the theorem regarding the maximality of the orthogonal group in the equiaffine group proved in (12). A. Schleiermacher and K. Strambach (12) proved a very interesting result regarding the maximaility of the group of orthogonal transformations and of that of Euclidean similarities inside certain groups of affine transformations. Although similar results have been proved earlier, this is the first time that the base field for the groups in question was not the field of real numbers, but an arbitrary Pythagorean field which admits only Archimedean orderings. They also state, as geometric significance of the result regarding the maximality of the group of Euclidean motions in the unimodular group over the reals, that there is no geometry between the classical Euclidean and the affine geometry. The aim of this note is to point out the exact geometric meaning of the positive part of the 2-dimensional part their theorem, in the case in which the underlying field is an Archimedean ordered Euclidean field. In this case their theorem states that: (1) the group G1 of Euclidean isometries is maximal in the group H1 of equiaffinities (affine transformations that preserve non-d irected area), and that (2) the group G2 of Euclidean similarities is maximal in the group H2 of affine transformations. The restriction to the 2-dimensional case is not essential but simplifies the presentation. The geometric counterpart of group-theoretic results in the spirit of the Erlanger Programm is given by Beth's theorem, as was emphasized by Buchi (1). Let Eu denote the class of Archimedean ordered Euclidean fields. Given that Eu is not an elementary class (i. e. cannot be axiomatized in first-order logic, as all of its models can be embedded in R, and thus cannot have models of cardinality > 2 ℵ0 , whereas, by the Lowenheim- Skolem theorem, first-order theories admitting infinite models have models of arbitrarily large cardinality), the logical interpretation one is bound to find for the above results will by necessity be one in a higher-order logic which is