The paper is devoted to the description of changes of the structure of the holomorphic automorphism group of a bounded domain in mathbb{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 mathbb{C}^n does not exceed n.