G-compactness and groups

Lascar described EKP as a composition of EL and the topological closure of EL ([1]). 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/GL. We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories.