Let $A\cong (\mathbb{Q}_p/\mathbb{Z}_p)^r$ and let $\Gamma\cong \mathbb{Z}_p$ act continuously on $A$, where we take $A$ with the discrete topology. I am trying to work out an exercise which asks to show that if $A^\Gamma$ is finite then $A/(\gamma-1)A$ is trivial, where $\gamma\in \Gamma$ is a topological generator. (Equivalently, the exercise is asking to show that if $H^0(\Gamma,A)$ is finite then $H^1(\Gamma,A)=0$.) Clearly there is an exact sequence $$ 0\rightarrow A^\Gamma\rightarrow A\xrightarrow{\gamma-1} A\rightarrow A/(\gamma-1)A\rightarrow 0, $$ so it suffices to show that if $\gamma-1$ has finite kernel then it is surjective. If $r=1$ then I don't think we even need the finiteness condition on $A^\Gamma$: since $\hom(\mathbb{Q}_p/\mathbb{Z}_p,\mathbb{Q}_p/\mathbb{Z}_p)\cong \mathbb{Z}_p$, the fact that $\gamma-1\neq 0$ implies that $\gamma-1$ is invertible. I'm having some trouble with the general case though... I know that $\hom(A,A)$ is isomorphic to the group of $r\times r$ matrices over $\mathbb{Z}_p$, so $\gamma-1$ corresponds to some matrix with $\mathbb{Z}_p$-coefficients, but I'm not quite sure what to do with that.
Note: Although this is technically not a question about Galois cohomology, I added the tag because $\Gamma$ is often the Galois group of a $\mathbb{Z}_p$-extension, and perhaps people that have dealt with these things might have some hints for me...