ABSTRACT
In this chapter we show how the idea of a regular language can be relevant in group theory. We define the central notion of this book, that of an automatic group (Definition 2.3.1), and find a characterization for such groups (Theorem 2.3.5). Roughly speaking, an automatic group is a finitely generated group for which one can check, by means of a finite state automaton, whether two words in a given presentation represent the same element or not, and whether or not the elements they represent differ by right multiplication by a single generator. The totality of these data-the generators and the automata--constitute an automatic structure for the group.
