Abstract
It is known Hrushovski and Pillay (Israel J Math 85:203–262, 1994) that a group G definable in the field \({\mathbb {Q}}_{p}\) of p-adic numbers is definably locally isomorphic to the group \(H({\mathbb {Q}}_{p})\) of p-adic points of a (connected) algebraic group H over \({\mathbb {Q}}_{p}\). 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 \({\mathbb {Q}}_{p}\) 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.
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Acknowledgements
Both authors would like to thank the Institut Henri Poincaré, Paris, for its hospitality and support during the trimester on model theory in early 2018 when this work was done. The second author would like to thank the IHES, Orsay, for its hospitality during the academic year 2017–2018. Both authors would like to thank Immi Halupczok for discussions. Finally, both authors would like to thank the referee for his or her comments which led to some positive changes in the paper. The introduction has been expanded to include comparisons with earlier results, and the proof of the main result, Proposition 2.3, has been improved.
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Anand Pillay: Supported by NSF Grants DMS-1360702, DMS 1665035, and DMS 1760413
Ningyuan Yao: Supported by NSFC Grant 11601090 and Shangai Puijan Program 16PJC018.
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Pillay, A., Yao, N. A note on groups definable in the p-adic field. Arch. Math. Logic 58, 1029–1034 (2019). https://doi.org/10.1007/s00153-019-00673-y
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DOI: https://doi.org/10.1007/s00153-019-00673-y