Let $G$ be a group $\mathbb { F }$ a field and $n$ a positive integer. A linear action of $G$ on $\mathbb { F } ^ { n }$ is a map $\alpha : G \times \mathbb { F } ^ { n } \rightarrow \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = \rho ( g ) v$ for some group homomorphism $\rho : G \rightarrow \mathrm { GL } _ { n } ( \mathbb { F } )$. Show that for every finite group $G$, there is an $n$ such that there is a linear action $\alpha$ of $G$ on $\mathbb { F } ^ { n }$ and such that there is a nonzero vector $v \in \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = v$ for all $g \in G$.
Let $G$ be a group $\mathbb { F }$ a field and $n$ a positive integer. A linear action of $G$ on $\mathbb { F } ^ { n }$ is a map $\alpha : G \times \mathbb { F } ^ { n } \rightarrow \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = \rho ( g ) v$ for some group homomorphism $\rho : G \rightarrow \mathrm { GL } _ { n } ( \mathbb { F } )$. Show that for every finite group $G$, there is an $n$ such that there is a linear action $\alpha$ of $G$ on $\mathbb { F } ^ { n }$ and such that there is a nonzero vector $v \in \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = v$ for all $g \in G$.