Kernel of a character ($\ker \chi =\{g:\chi (g)=\chi (1)\}$)

For a complex character $\chi$ of a finite group, its kernel is the kernel of an affording group representation. Averaging an inner product makes that representation unitary. If its degree is $d$, all eigenvalues of a group element have modulus one, so their sum equals $d$ only when every eigenvalue is one. Thus the displayed character-value criterion agrees with the representation kernel and defines a normal subgroup.