Let $V$ be a finite dimensional vector space over $\mathbb{F}_p$ with an action of the cyclic group $C_p$ of order $p$. We can now form the two vector spaces of invariants $V^{C_p}$ and co-invariants $V_{C_p}$.
Qustion: Do we always have $\dim{V^{C_p}=\dim V_{C_p}}?$
Just to say what I do know about this question:
This is true for the trivial representation and the regular representation and hence for any direct sum of them. Namely, for any permutation representation.
When working over a field of characteristic 0, the canonical norm map provides a natural isomorphism from the co-invariants to the invariants.
Over $\mathbb{F}_p$, the norm map is not an isomorphism and in fact, no natural isomorphism exists.
For arbitrary $p$-groups $G$ in place of $C_p$ this is false.
Well, $V^{C_p}$ and $V_{C_p}$ are the kernel and cokernel of $V\xrightarrow{g-1}V$, $g$ a generator of $C_p$, so they have the same dimension.