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ON REGULAR GROUPS AND FIELDS

Published online by Cambridge University Press:  18 August 2014

TOMASZ GOGACZ  and
KRZYSZTOF KRUPIŃSKI
TOMASZ GOGACZ
Affiliation:
INSTYTUT INFORMATYKI, UNIWERSYTET WROCŁAWSKI, UL. JOLIOT-CURIE 15, 50-383 WROCŁAW, POLANDE-mail: [email protected]
KRZYSZTOF KRUPIŃSKI
Affiliation:
INSTYTUT MATEMATYCZNY, UNIWERSYTET WROCŁAWSKI, PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW, POLANDE-mail: [email protected]
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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 nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1 with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable.

Keywords

MSCPrimary 03C60Secondary 12L1220A1520E0603C45minimalquasi-minimalgroupfieldregular typegenerically stable typeHNN-extension
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 3 , September 2014 , pp. 826 - 844
Copyright
Copyright © The Association for Symbolic Logic 2014 

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