I'm going over Marker's Model Theory. I have two questions in the "$\omega$-Stable groups" section.
In Lemma 7.1.12: $p\in S_1(G)$ and $\psi(v)$ defines $G^0$. He claims that there exists a $b\in G$ such that $\psi(b^{-1}v)\in p$. Why is that?
Why are there only a finite number of types of maximal Morley rank, that is of rank equal to the rank of the group?
For 1., notice that $p(v)\cup p(v')\vdash \psi(v'^{-1}v)$, so for some finite $p_0\subseteq p$ we have $p(v)\cup p_0(v')\vdash \psi(v'^{-1}v)$ and $p_0$ is of course satisfied in $G$.
For 2., if there are infinitely many points (types) of a given Cantor-Bendixson rank (Morley rank) at least $\alpha<\infty$ in a given clopen set (the set of types satisfying a formula, like the entire $S_1(G)$), then the rank of the set is at least $\alpha+1$. On the other hand, $\omega$-stability is equivalent to statement that the Morley rank is bounded.