$ \DeclareMathOperator{\RM}{RM} $
I have a problem with understanding part b) of Exercise 6.6.23 in David Marker's "Model Theory: An Introduction": ($ \mathbb M $ is the monster model.)
a) Suppose that $ \mathbb M $ is $ \omega $-stable, $ A , B \subseteq \mathbb M ^ n $ are definable, $ \RM ( A ) $ is finite and $ f : A \to B $ is a definable surjective map such that $ \RM \big( f ^ { - 1 } ( b ) \big) = k $ for all $ b \in B $. Show that $ \RM ( A ) \ge \RM ( B ) + k $. [Hint: Prove by induction on rank that $ \RM \big( f ^ { - 1 } ( X ) \big) \ge \RM ( X ) + k $ for all definable $ X \subseteq B $.]
b) Suppose that $ G $ is an $ \omega $-stable group of finite Morley rank and $ H \le G $ is an infinite definable subgroup. Show that $ \RM ( G ) \ge \RM ( H ) + \RM ( G / H ) $. In particular, $ \RM ( G ) > \RM ( G / H ) $.
c ) Show that b) is not true for all $ \omega $-stable groups. [Hint: Let $ K $ be a differentially closed field and consider the derivation $ \delta : K \to K $.]
First of all, doesn't $ H $ have to be a normal subgroup of $ G $ so that $ G / H $ can be defined? Or does $ G / H $ mean something other than the quotient group? And lastly, as far as I know, Morley rank is defined on formulas and definable subsets of the [monster] model. So, how is the question about the Morley rank of $ G / H $, which consists of cosets?
The algebra question: When $H$ is an arbitrary subgroup of $G$, $G/H$ often denotes the set of left cosets of $H$, $\{gH\mid g\in G\}$. When $H$ fails to be normal, we can't equip $G/H$ with a natural group structure, but we can equip it with the structure of a $G$-set: the action of $G$ on $G/H$ is the obvious one $g'\bullet (gH)= g'gH$.
The model theory question: While $G/H$ is not a definable set in $G$, it is an imaginary definable set, i.e. a definable set in $G^{\text{eq}}$. Precisely, suppose $\varphi_H(x)$ is the formula defining $H$. Then there is a definable equivalence relation $E(x,y)$ on $G$, given by $\exists z\, (\varphi_H(z)\land xz = y$), and the equivalence classes mod $E$ are exactly the left cosets of $H$. Since Morley rank makes sense in $G^{\text{eq}}$, we can define $\mathrm{RM}(G/H)$ to be the Morley rank of the set of $E$-classes in $G^{\text{eq}}$.