Inner product invariant under group action - meaning?

Question

My professor wrote a proposition where he defined an inner product on a $G$-module $V$. One of the premises was that the inner product was 'invariant under the action of the group', however he did not mention what this means. I assume that it either means that $\langle \boldsymbol{u},\boldsymbol{v}\rangle=\langle g\boldsymbol{u},g\boldsymbol{v}\rangle$, or more strictly that $\langle \boldsymbol{u},\boldsymbol{v}\rangle=\langle g\boldsymbol{u},\boldsymbol{v}\rangle=\langle\boldsymbol{u},g\boldsymbol{v}\rangle$, for any $g\in G$ and $\boldsymbol{u},\boldsymbol{v}\in V$, but this is just a guess.

Answer 1

It means the first condition, i.e. $\langle u,v\rangle = \langle gu,gv \rangle$ for all $g\in G$ and $u,v\in V$. Notice that the second condition would imply that $gv=v$ for all $g\in G$ and $v\in V$ which is only true if $V$ is a trivial representation.