Let $G$ be a finite group and $$ 0 \to P \stackrel{\phi}{\to} M \stackrel{\psi}{\to} N \to 0 $$ be an exact sequence of $G$-modules. In Appendix B of Silverman's Arithmetic of Elliptic Curves, it is said that taking the $G$-invariants gives an exact sequence $$ 0 \to P^G \stackrel{\phi}{\to} M^G \stackrel{\psi}{\to} N^G $$ but $\psi: M^G \to N^G$ does not necessarily have to be surjective.
Could you give me an example for such a case? I would like to understand this better but find these definitions about $G$-modules too abstract to begin with.
Hint:
Look at the sequence
$$ 0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2 \to 0 $$
as $\mathbb{Z}/2$-modules. Here the action on each module is by $x \mapsto -x$ (note this makes the action on $\mathbb{Z}/2$ trivial). Notice this means the action commutes with each of the module homs as well, so this really is a sequence of $\mathbb{Z}/2$-modules.
Now take $\mathbb{Z}/2$ invariants. Do you see what you get in each case? I'll include the answer under the fold, but you should really try to compute it yourself first!
As an aside, one can show the $G$-invariant functor is naturally isomorphic to the functor $\text{Hom}_{\mathbb{Z}G}(\mathbb{Z}, -)$. This might give some intuition for why this functor is left exact, but not exact in general. In fact, just like we can measure the failure of exactness of $\text{Hom}$ by looking at cohomology, there is a notion of group cohomology which measures the failure of exactness for this functor.
I hope this helps ^_^