Definable Groups for Dependent and 2-Dependent Theories

Source abstract

Let $T$ be a (first order complete) dependent theory, ${\gC}$ a $\bar\kappa$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $< \bar\kappa$ type-definable subsets there is a minimal one, i.e. their intersection has bounded index. See history in \cite{Sh:876}. We then deal with definable groups for 2-dependent theories, a wider class of first order theories proving that for many pairs $(M,N)$ of models, the minimal bounded subgroup definable over $M \cup N$ is the intersection of the minimal ones for $M$ and for $N$.