Definable groups for dependent and 2-dependent theories

Abstract

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ 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. In fact, the bound is $\leq 2^{|T|}$. We then deal with 2-dependent theories, a wider class of first order theories.