Recently, I've been reading Shatz's book, profinite groups, arithmetic, and geometry. Let $G$ be a finite $p$-group and $A$ a finite $G$-module such that $pA=(0)$. In the proof of Theorem 19 (p.82), Shatz defines the Poincare polynomial of A by $\chi_{A}(t) = \sum\limits_{i=0}^{\infty} c_n (A) t^n $ where $c_n (A)$ is the dimension of $I_G^nA/I_G^{n+1}A$ as vector space over $\mathbb{Z} / p \mathbb{Z}$. He says that because G is solvable and $I_G/I_G^2$ is isomorphic to $G^{ab}$, $I_G^mA/I_G^{m+1}A = (0)$ for large m and so $\chi_{A}(t)$ is really a polynomial. Could anyone tell me why? I'm guessing that $I_G^n/I_G^{n+1}$ has something to do with the commutator series of $G$, i.e., $G^{(0)} = G$ and $G^{(i+1)} = [G^i,G^i]$. But I don't know how exactly.
Thank you.