Projective Groups over Rings

Abstract

In this paper, Mäurer's theorems characterizing certain subgroups of the projective group PGL 2 K over a field K are generalized to the case of rings. Key Words: projective lines over rings; projective groups over rings; permutation groups.In [9] and [10] Mäurer modifies the famous theorem due to Tits that characterizes the permutation groups PGL 2 K over commutative fields among all sharply triply transitive permutation groups. Mäurer's results are based upon the concept of symmetry instead of transitivity. The aim of the present paper is to carry over Mäurer's theorems to the case of a large class of rings, including the commutative algebras R over a field K such that n = dim K R < ∞ and K > 5n.For the reader's convenience, in Section 1 we recall the theorems by Tits and Mäurer, respectively. Moreover, we introduce the projective line over a ring and groups acting on it. We state a generalization of Tits's theorem for projective groups over rings due to Herzer.In Section 2 we find appropriate generalizations of the conditions used by Mäurer for his characterization of certain subgroups of PGL 2 K . The coordinatization results we are aiming at will require them as axioms. We are concerned with a permutation group acting on a point set and leaving invariant a certain relation on called "distant."1 The results of this paper are part of the author's Habilitationsschrift [3]. I thank the referee for his/her valuable suggestions on how to shorten Section 6. 266

Divisible designs, Laguerre geometry, and beyond

“…It contains a wealth of bibliographical data. Characterisations of projective groups PGL 2 (R), where R is a ring, are given in [23], [28] and [69]. See also [40,Chapter 6].…”

“…alternative algebras or Jordan systems) in order to obtain a kind of "chain geometry". We refer to [11], [12], [18], [19], [20], [21], [28], [40,Chapter 3], [41], and [68].…”

Chain spaces with many reflections

“…In [5] we investigated pairs ( , (P, )), where P is a non-empty set, is a symmetric, anti-reflexive relation on P, called distant, and is a group of permutations of P leaving invariant. In particular, we made use of a distinguished subsetÎ ⊆ consisting of all involutions satisfying two additional properties (I1), (I2) (see [5], 2.1).…”

“…In particular, we made use of a distinguished subsetÎ ⊆ consisting of all involutions satisfying two additional properties (I1), (I2) (see [5], 2.1).…”

“…The aim of this paper is to apply the results of [5] to these geometries. So consider an arbitrary ( , ), and letÎ ⊆ be the distinguished set of involutions as described above; we now call the elements ofÎ reflections.…”