A representation $\rho:G\rightarrow GL(V)$ of a finite group is faithful if and only if it is injective. Prove that $\rho$ is faithful if and only if the only $g\in G$ with $\chi(g)=\dim V$ is $g =1$.
($\Rightarrow$): Suppose $\rho$ is injective. Then whenever $\rho(g).v=v$, $g=1$. Now let $g\in G$ be any element in $G$ such that $\chi(g)=\dim V$
How to continue from here?
Best Answer
You have to use the following fact: the trace of a matrix $A$ whose order is finite is $\dim(V)$ if and only if $A=Id_V$ (see the reference for a complete proof).
https://yutsumura.com/finite-order-matrix-and-its-trace/
Since $G$ is finite, the order of the elements of its image are finite. This implies that $trace(\rho(g))=dim(V)$ is equivalent to $\rho(g)=Id_V$.