Finite cardinality modules over $R=\Lambda(\Gamma)\cong\mathbb Z_p[[T]]$ Let $Q$ be a finite cardinality module over $R$. Let $\Gamma\cong\mathbb Z_p$, ($\Gamma$ is written multiplicatively, and in $\mathbb Z_p[[T]]$ corresponds to $(1+T)^{\mathbb Z_p}$). We have the following obvious inclusions among invariant modules:
$$Q^{\Gamma} \subseteq Q^{\Gamma^p} \subseteq Q^{\Gamma^{p^2}} \subseteq \ldots $$
Is it true that this chain will stabilise and eventually equal $Q$?
This result seems to be used multiple times in Sujatha & Coates's book "Cyclotomic fields and Zeta values".
If yes, how can one see it?
Thanks!!

The action of $R$ on $Q$ gives a homomorphism from $\mathbb{Z}_p$ to the group $A$ of automorphisms of $Q$.  Since $Q$ is finite, $A$ is finite, so this homomorphism factors through the quotient $\mathbb{Z}_p\to\mathbb{Z}/p^n$ for some $n$.  For that value of $n$, then, $Q^{\Gamma^{p^n}}=Q$.
(Note that the fact that the sequence must stabilize at at some value is trivial, since any nested sequence of subsets of a finite set eventually stabilizes.)
