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.