What the's contradiction in showing the regular representation is indecomposable in characteristic $p$? Suppose $G$ is a nontrivial $p$ group, and $F$ is a field of characteristic $p$. The group ring $FG$ is a module over itself affording the regular representation $g\cdot g_i=gg_i$. 
Why is $FG$ indecomposable? I read a proof saying that if $FG=M_1\oplus M_2$ is decomposable, then each $M_i$ contains the trivial subrepresentation $\tau_i$, so $FG$ has $\tau_1\oplus \tau_2$ as a subrepresentation. 
But the subspace of $FG$ fixed by all of $G$ is the set of $F$-multiples of $\sum_{g\in G}g$, which has dimension $1$, and apparently this is a contradiction. I understand all the steps stated, but why is this a contradiction?

The contradiction is because in the first case $FG$ has two district trivial submodules, whereas in fact there is a unique trivial submodule spanned by the sum you mention.
