G-compactness and groups

Lascar described E(KP) as a composition of E(L) and the topological closure of EL (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.