Let $G$ be a group and $p$ a prime which divides $|G|$. Let $F$ be a field of characteristic $p$. Let $\epsilon\in F[G]$ - the group algebra - be the sum all elements of $G$.
How to show that the $F[G]$-submodule $F[G]\epsilon$ is not a direct summand of $F[G]$?
The only thing I realize is that in this case $F[G]\epsilon$ is a submodule of the augmentation - the module spanned by elements $\Sigma a_gg$ where $\Sigma a_g=0$. Is this relevant?