A generalization of Jordan groups

Abstract

Abstract For r to be a Jordan set in a group (G, 0) in the usual definition (see p. 73 in the article by Macpherson in this volume) the pointwise stabilizer, G(o,r), must be transitive on r. In this paper we relax this condition and demand that the setwise stabilizer, G(r), be transitive on r. But we also add as assumptions two properties which hold for Jordan sets in the usual definition; namely that the union of any chain of the (new) Jordan sets be a (new) Jordan set, and that the union of any two non-disjoint (new) Jordan sets be a (new) Jordan set. Although infinite simply primitive Jordan groups in the new sense lead to the same G-invariant structures as the old ones of Adeleke and Neumann (in press b), there are simply primitive groups which are Jordan in the new sense but not in the usual sense. Immediate examples are the pathological groups on linear orders described by Glass (1981, Chapter 6, Section 1.10). We make remarks in Section 4 below about the construction of such examples, and it follows easily that there are more examples of the same type built from semilinear orders and C-relations. Apart from including more groups, the result below should be useful in any eventual classification of general infinite simply primitive groups according to their invariant structures. The new Jordan sets and Jordan groups shall be called c-Jordan sets and c-Jordan groups respectively (‘c-Jordan’ as in ‘Camille Jordan’). The proofs of Adeleke and Neumann (in pressb, to appear) need several adjustments to handle the present context.