Group actions and invariant subspaces

Question

Let $G:= S_3$ (So we have $\{e,(12),(13),(23),(123),(132)\}$

And it acts on some three dimensional vector space via permutation.

Let $U$ be the subspace $\{(a,b,c) \mid a+b+c=0\}$

I must show $\nexists W \subsetneq U$ s.t. $gW \subseteq W$ $\forall g \in S_3$, i.e., $U$ has no invariant subspace.

First can I think of $U$ as the span of $\{(-1,1,0),(0,1,-1)\}$

Then I must generalize to $S_n$ acting on an $n$ dimensional vector space.

Any hints or tips on how to go about this? I know the dimension of this subspace is one less than dimension of our vector space? Since we are putting one restriction on it? So dim $U$=$2$.

Answer 1

You're right that $U$ has dimension $2$. The only proper (nontrivial) subspaces are then one-dimensional. Thus if such a subspace existed, it would be generated by some $v$. But then $gv = \lambda_g v$ for all $g$. Can you take it from here?