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Show the existence of a 1-dimensional invariant subspace for any 5-dimensional complex representation of the group $A_4$, where $A_4$ is the alternating group of degree 4.

Any hints?

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    I think you could, for the sake of completeness, include the definition of the group $A_4$.2012-05-09
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    Over what field?2012-05-09
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    I got this problem already. Consider the character table for $A_4$, which contains three 1-dim irreduciable characters and one 3-dim irreduciable character. Since 5=1+1+1+1+1 or 5=1+1+3, so a 5-dim represtation must contains a 1-dim invariant subspace.2012-05-09
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    @ChrisEagle Over $\mathbb{C}$2012-05-09
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    @CC_Azusa Please consider writing the answer up in the answer box, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-10-04

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