Automorphism groups over a hyperimaginary

In this paper we study the Lascar group over a hyperimaginary 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 simplifies 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-definable. On the one hand, we correct errors appeared in the book, "Simplicity Theory" [6, 5.1.14-15] and produce a counterexample. On the other, we extend Newelski's Theorem in "The diameter of a Lascar strong type" [12] that `a 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 raised in "The relativized Lascar groups, type-amalgamations, and algebraicity" [4, 2.11], which is even a new result in the real context.