$\newcommand{\GL}{\operatorname{GL}} \newcommand{\ip}[1]{\langle #1 \rangle} \newcommand{\bip}[1]{\Big \langle #1 \Big \rangle} \newcommand{\set}[1]{\left\{ #1 \right\}}$ Let $V$ be an inner product space over the field $K$ with inner product $\ip{\cdot,\cdot}$. Let $G := \set{g_i}_{i=1}^n$ be a finite subgroup of $\GL(V)$. Define the mapping $\ip{\cdot,\cdot}_G : V \times V \to K$ by
$$\ip{x,y}_G = \frac{1}{n} \sum_{i=1}^n \bip{g_i(x),g_i(y)}$$
I have already shown that $\ip{\cdot,\cdot}_G$ is an inner product on $V$. I have to show two more things:
- $\forall g \in G$ and $\forall x,y \in V$, this holds:
$$\bip{g(x),g(y)}_G = \ip{x,y}_G$$
- $\exists u \in \GL(V)$ such that, $\forall x,y \in V$, this holds:
$$\bip{u(x),u(y)}_G = \ip{x,y}$$
However, I'm not really sure how to do either of these.
For the first one, it is noteworthy that
$$\Big \langle g(x),g(y) \Big \rangle_G = \frac 1 n \sum_{i=1}^n \Big \langle (g_i \circ g)(x),(g_i \circ g)(y) \Big \rangle$$
Since $G$ is a subgroup of $\GL(V)$, then, $g_i \circ g \in G \; \forall i$. This seems to me to imply that, if the $g \in G$ "shuffles around" or "permutes" the $g_i$ in the summation somehow (but otherwise doesn't change them), then the desired result essentially follows. However, I'm not fully sure how to rigorously argue this, or if it even is necessarily true.
For the second one, I do notice that $\ip{\cdot,\cdot}_G$ has this structure sort of like an "average" of sorts. It would be very convenient for me if the $u \in \GL(V)$ in question was such that $(g_i \circ u)(v) = v$ $\forall v \in V$. Then you would just have the average of $n \ip{x,y}$. This suggests something to do with inverses, which are guaranteed to exist in $\GL(V)$, but inverses are also unique, so it doesn't feel like any one $u \in \GL(V)$ can achieve quite this angle of approach. So I'm lost there, too.
Is there any sort of way forward that you can help me with? A nudge forward or hint? I've been stuck for a while on this problem, so any help would be appreciated!
(Note: This is for a homework assignment, so I would prefer only hints or nudges in the right direction, as opposed to full solutions.)
Update: Thanks to a friend on Discord and Chrystomath in the comments, it's clear on me that the first problem essentially hinges on the fact that $gG=G$ (or $Gg=G$ had I presented it correctly). Not sure how I didn't notice that before.