G-compactness and groups

Lascar described E KP as a composition of E L and the topological closure of E L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non- G -compact theory, we consider the following example. Assume G is a group definable in a structure M . We define a structure M ′ consisting of M and X as two sorts, where X is an affine copy of G and in M ′ we have the structure of M and the action of G on X . We prove that the Lascar group of M ′ is a semi-direct product of the Lascar group of M and G / G L . We discuss the relationship between G -compactness of M and M ′. This example may yield new examples of non- G -compact theories.