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

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 :)

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• You might start by seeing what $g\cdot N_G$ is for an arbitrary $g\in G$. From there, maybe you can describe what $(N_G)$ is? – David Hill Feb 5 '16 at 15:51
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.