  The paper is devoted to the description of changes of the structure of the
holomorphic automorphism group of a bounded domain in ${\Bbb C}^n$ under small
perturbation of this domain in the Hausdorff metric. We consider a number of
examples when an arbitrary small perturbation can lead to a domain with a
larger group, present theorems concerning upper semicontinuity property of some
invariants of automorphism groups. We also prove that the dimension of an
abelian subgroup of the automorphism group of a bounded domain in ${\Bbb C}^n$
does not exceed $n$.
