Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$.

Question

I am trying to solve this Representation Theory question:

Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$.

Any help would be highly beneficial. Thanks :)

Answer 1

Show that the left ideal $F[G]N_G=FN_G$, and you will have shown that it is $1$-dimensional as an $F$-subspace, and that is necessarily a simple left module.