Automorphism groups of numbered sets (Q1099169)

A surjection \(\alpha\) : \(N\to A\) is called a numeration of A, and in this paper \b{A}\(=(A,\alpha)\) is called a numbered set. Let Aut(\b{A}) denote the group of automorphisms of \b{A} in the category of all numbered sets. The author shows that the class of all groups of the form Aut(\b{A}) coincides up to isomorphism with the class of all countable groups. The technique of proof also establishes a result independently due to Morozov: the class of all groups of automorphisms of finite numbered sets coincides up to isomorphism with the class of all finite groups.