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A note on groups definable in the p-adic field

A note on groups definable in the p-adic field It is known Hrushovski and Pillay (Israel J Math 85:203–262, 1994) that a group G definable in the field $${\mathbb {Q}}_{p}$$ Q p of p-adic numbers is definably locally isomorphic to the group $$H({\mathbb {Q}}_{p})$$ H ( Q p ) of p-adic points of a (connected) algebraic group H over $${\mathbb {Q}}_{p}$$ Q p . We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in $${\mathbb {Q}}_{p}$$ Q p is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the more general context of geometric structures. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

A note on groups definable in the p-adic field

Archive for Mathematical Logic , Volume 58 (8) – Apr 29, 2019

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References (12)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-019-00673-y
Publisher site
See Article on Publisher Site

Abstract

It is known Hrushovski and Pillay (Israel J Math 85:203–262, 1994) that a group G definable in the field $${\mathbb {Q}}_{p}$$ Q p of p-adic numbers is definably locally isomorphic to the group $$H({\mathbb {Q}}_{p})$$ H ( Q p ) of p-adic points of a (connected) algebraic group H over $${\mathbb {Q}}_{p}$$ Q p . We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in $${\mathbb {Q}}_{p}$$ Q p is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the more general context of geometric structures.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Apr 29, 2019

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