## Abstract

In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of stp(ac) is abelian, and the underlying theory is G-compact, then the Lascar groups of stp(ac) and of stp(a) are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of stp(ac) is abelian, nor the assumption of c being finite can be removed.

Original language | English |
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Pages (from-to) | 531-557 |

Number of pages | 27 |

Journal | Journal of Symbolic Logic |

Volume | 86 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2021 Jun 1 |

### Bibliographical note

Funding Information:Let b,b′ |= pn where (b,b′) is an endpoint pair of a 1-shell s′ of pn such that B(s′) = 7 in M1,n. Let α′ be a minimal 2-chain with an RN-pattern having the 1-shell boundary s′ of length 7. Note that we can take such a 2-chain because pn is a Lascar type. Consider a = (x0,b),a′ = (x6,b′) |= q. Let s′′ be a 1-shell induced from s and s′ with an endpoint pair (a,a′). Consider a 2-chain α′′ in q induced from α and α′, which does not a Lascar pattern and has length 7. We claim that α′′ is minimal. Since B(s) ≤ 7 and B(s′) = 7, we have that B(s′′) = 7 and so α′′ is minimal. By the construction of α′′, it has an RN-pattern. Therefore, the minimal 2-chain α′′ has an RN-pattern but is not equivalent to a Lascar pattern 2-chain. ⊣ Acknowledgments. The first author was supported by European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 705410. The second and fourth authors were supported by an NRF of Korea grant 2018R1D1A1A02085584 and Samsung Science Technology Foundation under Project Number SSTF-BA1301-03, respectively.

Publisher Copyright:

© The Author(s), 2021. Published by Cambridge University Press.

## All Science Journal Classification (ASJC) codes

- Philosophy
- Logic