Relativized Galois groups of first order theories over a hyperimaginary

Abstract

We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $\Sigma$. We introduce the notion of a Lascar tuple for $\Sigma$ and by considering the space of types over a Lascar tuple for $\Sigma$, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type $\Sigma$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types.