3 tag p-groups
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# Representation theory of a finite p-group over a field of characteristic p: dim of invariants =1 => dim of coinvariants = 1?

I am trying to understand the proof of Proposition 4 in S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. Volume 39 (1970), 141-148. The PDF is available here: http://projecteuclid.org/euclid.nmj/1118798052

It appears that the following result is used, but I'm afraid that I don't quite see the proof. Let $k$ be a field of characteristic $p$ and let $G$ be a finite $p$-group. Let $W$ be a left $k[G]$-module such that $\dim_k W = |G|$. Suppose that $\dim_k W^{G} = 1$. Then $\dim_k W_G = 1$. Here $W^G$ denotes invariants and $W_G$ denotes coinvariants.

I have to admit that I know relatively little about representation theory in characteristic $p$. One idea would be to consider the dual representation $\hat{W}$, but I only got so far: $\dim_k W^{G} = 1 \implies W$ is indecomposable $\implies \hat{W}$ is indecomposable. But maybe this is not the right approach. If I could show that the Tate cohomology groups of $W$ vanish, then of course the desired result drops out, but I think this is rather strong medicine.

Is anyone able to give a proof of the above claim? I suspect the solution is fairly easy, but I just don't see it at the moment.