Let $V$ be an $(N+1)$-dimensional vector space with an action of the symmetric group $S\_n$, such that $V$ is an irreducible $S\_n$ -module. 


Let $\{p\_1,...,p\_h\}\in \mathbb{P}(V)$ be $h\geq N+2$ points such that $S\_n$ acts transitively on $\{p\_1,...,p\_h\}$. 


Is it true then that there exist $N+2$ points $p\_{i\_1},...,p\_{i\_{N+2}}\in \{p\_1,...,p\_h\}$ that are in linear general position in $\mathbb{P}(V)$ ?



Irreducible $S_n$-modules and $S_n$-actions on projective spaces
**EDIT**: Actually, I made this way too difficult. 


Points in projective space are in general linear position if the corresponding lines are the multiples of a basis of $V$. Thus, the question is just if the lines corresponding to $p\_1,\dots, p\_h$ span the vector space $V$ (since in this case, some subset will correspond to a basis). The span of these lines is invariant and non-zero, and so must be all of $V$ by its irreducibility. 


