Let $\varphi:G\to GL(V)$ be a representation of a finite group $G$. Define the subspace $$V^G=\{v\in V\mid\text{for all } g\in G,gv=v\}.$$ How can I show that $V^G$ is a $G$-invariant subspace.
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2There is nothing to show. It is just by definition of $G$-invariant and $V^G$. – Dietrich Burde May 01 '17 at 16:04
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1Think what you want to show: "If $v\in V^G$ and $g\in G$, then $gv\in V^G$" – Hagen von Eitzen May 01 '17 at 16:09