Past exam of the algebra course of the University of Oxford / 2010 / B / 6 / a / Solution

A complex character of a representation of a finite group $G$ is the functionwhere $\rho :G\to \mathrm{GL}(V)$ is a finite-dimensional complex group representation. The character is a class function because trace is unchanged by conjugation. Its degree is $\chi (1)=\dim V$.