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**NonCommAlg** suppose $\displaystyle FG=V \oplus W$ and let $\displaystyle e=\sum_{g \in G} g.$ see that $\displaystyle ge=e$ for all $\displaystyle g \in G$ and thus $\displaystyle ve=0,$ for all $\displaystyle v \in V.$ we have $\displaystyle 1=v+w,$ for some $\displaystyle v \in V, \ w \in W$ and so $\displaystyle e=we.$ now suppose that $\displaystyle e \in V.$

then since $\displaystyle V,W$ are $\displaystyle FG$ submodules, we'll have $\displaystyle e=we \in V \cap W = \{0 \}, \color{red}^*$ which is nonsense. hence $\displaystyle e \notin V,$ which means that $\displaystyle |G|,$ as an element of $\displaystyle F,$ is non-zero. Q.E.D.

$\displaystyle \color{red}*$: note that $\displaystyle V$ is an ideal of $\displaystyle FG,$ i.e. it's both left and right $\displaystyle FG$ sumodule. so if we assume that $\displaystyle W$ is a right $\displaystyle FG$ submodule and $\displaystyle e \in V,$ then $\displaystyle we \in V \cap W = \{ 0 \}.$