# 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]

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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 :)

## marked as duplicate by rschwieb abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Feb 5 '16 at 16:22

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

## 1 Answer

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.