The model theory of unitriangular groups

Abstract

The model theory of groups of unitriangular matrices over rings is studied. An important tool in these studies is a new notion of a quasiunitriangular group. The models of the theory of all unitriangular groups (of fixed nilpotency class) are algebraically characterized; it turns out that all they are quasiunitriangular groups. It is proved that if R and S are domains or commutative associative rings then two quasiunitriangular groups over R and S are isomorphic only if R and S are isomorphic or antiisomorphic. This algebraic result is new even for ordinary unitriangular groups. The groups elementarily equivalent to a single unitriangular group UT n(R) are studied. If R is a skew field, they are of the form UT n ( S), for some S ≡ R. In general, the situation is not so nice. Examples are constructed demonstrating that such a group need not be a unitriangular group over some ring; moreover, there are rings P and R such that UT n(P) ≡ UT n(R) , but UT n(P) cannot be represented in the form UT n(S) for S ≡ R. We also study the number of models in a power of the theory of a unitriangular group. In particular, we prove that, for any communicative associative ring R and any infinite power λ, I(λ, R) = I(λ, UT n ( R)). We construct an associative ring such that I(ℵ 1, R) = 3 and I(ℵ 1 UT n(R)) = 2 . We also study models of the theory of UT n(R) in the case of categorical R.