Abstract:
Two ways of describing a group are considered. 1. A group is finite-automaton
presentable if its elements can be represented by strings over a finite
alphabet, in such a way that the set of representing strings and the group operation
can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely
axiomatizable if there is a description consisting of a single first-order sentence, together
with the information that the group is finitely generated. In the first part
of the paper we survey examples of FA-presentable groups, but also discuss theorems
restricting this class. In the second part, we give examples of quasi-finitely
axiomatizable groups, consider the algebraic content of the notion, and compare it
to the notion of a group which is a prime model. We also show that if a structure
is bi-interpretable in parameters with the ring of integers, then it is prime and
quasi-finitely axiomatizable.