Do you know what a group representation is? Do you know what the trace of a group representation is?
Solution
A finite-dimensional representation of a group \(G\) over a field \(F\) is a homomorphism \[ \rho:G\to\operatorname{GL}(V) \] for some finite-dimensional \(F\)-vector space \(V\).
The trace attached to the representation is its character \[ \chi_\rho:G\to F, \qquad \chi_\rho(g)=\operatorname{tr}(\rho(g)). \]
1. The character is constant on conjugacy classes.
For \(g,h\in G\), \[ \rho(hgh^{-1})=\rho(h)\rho(g)\rho(h)^{-1}. \] Trace is invariant under similarity, so \[ \chi_\rho(hgh^{-1})=\chi_\rho(g). \]
2. Isomorphic representations have the same character.
If \(\rho_1\) and \(\rho_2\) are isomorphic, then for some invertible \(T\), \[ \rho_2(g)=T\rho_1(g)T^{-1} \] for every \(g\). Again trace is similarity-invariant.
Thus \(\chi_\rho\) is a class function that depends only on the isomorphism class of \(\rho\); for a finite group over \(\CC\), it determines \(\rho\) up to isomorphism.