Groupes equationnellement minimaux

Abstract

A well-known definition in Model Theory states that an infinite structure M is minimal if any subset of M which is definable with parameters in M is either finite or cofinite. The starting point of the study of the groups of finite Morley rank is a theorem due to Reineke, saying that a minimal group is commutative. We introduce here a weaker notion, easily understandable by non-logicians : an infinite group G is equationally minimal if every equation in one variable, with coefficients in G, has in G either a finite or a cofinite number of solutions. Non-commutative equationally minimal groups probably exists: they will be non locally finite groups of exponent p, for a sufficiently large prime number p.