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January, 2023 Automorphism groups over a hyperimaginary
Byunghan KIM, Hyoyoon LEE
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J. Math. Soc. Japan 75(1): 21-49 (January, 2023). DOI: 10.2969/jmsj/87138713


In this paper we study the Lascar group over a hyperimaginary $\boldsymbol{e}$. We verify that various results about the group over a real set still hold when the set is replaced by $\boldsymbol{e}$. First of all, there is no written proof in the available literature that the group over $\boldsymbol{e}$ is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book written by the first author and produce a counterexample. On the other hand, we extend Newelski's theorem that ‘a G-compact theory over a set has a uniform bound for the Lascar distances’ to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context.

Funding Statement

The authors were supported by NRF of Korea grants 2018R1D1A1A02085584 and 2021R1A2C1009639.


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Byunghan KIM. Hyoyoon LEE. "Automorphism groups over a hyperimaginary." J. Math. Soc. Japan 75 (1) 21 - 49, January, 2023.


Received: 15 June 2021; Published: January, 2023
First available in Project Euclid: 16 June 2022

Digital Object Identifier: 10.2969/jmsj/87138713

Primary: 03C60
Secondary: 54H11

Keywords: G-compactness , hyperimaginary , Lascar group , strong types

Rights: Copyright ©2023 Mathematical Society of Japan


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Vol.75 • No. 1 • January, 2023
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