I am trying to work on some questions back from my uni days, and one has gotten the better of me at the moment!
Let $G$ be a finite group and $V, W$ finite-dimensional $\mathbb{C}G$-modules. Let $L:V\rightarrow W$ be a linear map and define $p:V\rightarrow W$ by $p(x)=\sum_{g\in G}g^{-1}L(g(x))$.
Prove that $p$ is a homomorphism of $\mathbb{C}G$-modules.
I have proven that $p$ is a linear map, but I geT stuck when trying to show that $p(h.x)=h.p(x)$. These are my steps thus far.
\begin{align} p(h.x) &= \sum_{g\in G} g^{-1}L(g(h.x)) \\ &= \sum_{g\in G} g^{-1}L((gh).x) \end{align}
I was pointed in the direction of using ${(gh).x:g\in G} ={g.x : g\in G}$ since ${gh:g\in G}=G$. But this isn't true, is it? Because ${gh:g\in G}$ would give a right coset $Gh$ instead of the whole group, right? And since $L$ isn't necessarily a $\mathbb{C}G$-homomorphism, we can't say that $L((gh).x)=(gh).L(x)$.
If anyone could help I would be extremely grateful!
Thanks,
Andy.
You have \begin{align*} p(h.x)&=\sum_gg^{-1}.L(g.(h.x))\\&=\sum_g(hh^{-1}g^{-1}).L((gh).x)\\ &=\sum_gh.[(gh)^{-1}.L((gh).x)]\\ &=h.\left(\sum_g(gh)^{-1}.L((gh).x)\right)\\ &=h.p(x) \end{align*} where the last equality follows from the fact that $$\sum_g(gh)^{-1}.L((gh).x=\sum_{gh^{-1}}g^{-1}.L(g.x)=\sum_{g}g^{-1}.L(g.x)=p(x).$$