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 amalgama-tion. The main result of this paper is that if c is a finite tuple alge-braic 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 com-pact 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 theo-retical 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 G-compactness can be removed.
|Publication status||Published - 2020 Apr 23|
All Science Journal Classification (ASJC) codes