 
Summary: SPECIAL GROUPS WHOSE ISOMETRY RELATION IS A
FINITE UNION OF COSETS
VINCENT ASTIER
Abstract. 0stable 0categorical linked quaternionic mappings are studied
and are shown to correspond (in some sense) to special groups which are 0
stable, 0categorical, satisfy AP(3) and have finite 2symbol length. They
are also related to special groups whose isometry relation is a finite union of
cosets, which are then considered on their own, as well as their links with
pseudofinite, profinite and weakly normal special groups.
The algebraic theory of quadratic forms is naturally divided into the reduced
theory of quadratic forms (corresponding to the theory of quadratic forms over
formally real Pythagorean fields) and the non(necessarily) reduced theory. The
former, with its links with the theory of orderings is much more developed, and a
striking example of this is Marshall's classification of spaces of orderings of finite
chain length ([20]). In the language of other axiomatisations of the algebraic theory
of quadratic forms, it tells us that Witt rings, or special groups, or linked quater
nionic mappings that are reduced and of finite chain length are completely classified.
There is no corresponding result for the nonreduced theory. However, reduced as
well as nonreduced special groups and linked quaternionic mappings are models of
firstorder theories, and Marshall's classification tells us that stable reduced special
