If the invariants are 0, then are the coinvariants also trivial? [closed]

Question

Let $G$ be a finite group and $M$ a $G$-module. Suppose that $M^G=0$. The group of coinvariants $M_G$ is the largest quotient of $M$ on which $G$ acts trivially. Is $M_G$ also trivial? It seems like it would be the case, but I don't know a proof.

Answer 1

This is not true in general.

The permutation module $P$ for the simple group ${\rm PSL}(2,5)$ (which is isomorphic to $A_5$) in its natural action on $6$ points over the fields of order $2$ is uniserial with submodules $P_1$ and $P_5$ of dimensions $1$ and $5$.

Then $M := P/P_1$ (with dimension $5$) has $M^G=0$ but $|M_G|=2$.