Submission date: 16 November 2012.

Abstract:

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let K be a regular field which is not generically stable and let p be its global generic type. We observe that if K has a finite extension L of degree n, then p^{(n)} has unbounded orbit under the action of the multiplicative group of L.
Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique non-trivial conjugacy class, and we notice that a classical group with one non-trivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then we construct a group of cardinality ω_1 with only one non-trivial conjugacy class and such that the centralizers of all non-trivial

Mathematics Subject Classification: 03C60, 12L12, 20A15, 20E06, 03C45

Keywords and phrases:

Full text arXiv 1211.3852: pdf, ps.