Introduction to the Lascar Group

The aim of this article is to give a short introduction to the Lascar Galois group GalL(T ) of a complete first order theory T . We prove that GalL(T ) is a quasicompact topological group in section 5. GalL(T ) has two canonical normal closed subgroups: Γ1(T ), the topological closure of the identity, and GalL(T ), the connected component. In section 6 we characterize these two groups by the way they act on bounded hyperimaginaries. In the last section we give examples which show that every compact group occurs as a Lascar Galois group and an example in which Γ1(T ) is non–trivial. None of the results, except possibly Corollary 26, are new, but some technical lemmas and proofs are. In particular, the treatment of the topology of GalL(T ) in sections 4 and 5 avoids ultraproducts, by which the topology was originally defined in [6]. Most of the theory expounded here was taken from that article, and the more recent [7], [4] and [2].