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This question already has an answer here:

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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marked as duplicate by rschwieb abstract-algebra Feb 5 '16 at 16:22

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ 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? $\endgroup$ – David Hill Feb 5 '16 at 15:51
  • $\begingroup$ Please try out the site's search function first, next time. I think you would have found the linked duplicate. $\endgroup$ – rschwieb Feb 5 '16 at 16:23
  • $\begingroup$ @rschwieb thank you for pointing this out - I am new to the stackexchange. $\endgroup$ – The Lost Unicorn Feb 6 '16 at 13:42
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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.

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