Automorphism groups over a hyperimaginary

In this paper we study the Lascar group over a hyperimag- inary e. We verify that various results about the group over a real set still hold when the set is replaced by e. First of all, there is no written proof in the available literature that the group over e is a topological group. We present an expository style proof of the fact, which even simpli_es existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-de_nable. On the one hand, we correct errors appeared in the book written by the _rst author and produce a counterexample. On the other hand, we extend Newelski's theorem that à G-compact theory over a set has a uniform bound for the Lascar distances' to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context.