(At time 0:03 ) Could you please explain to me how the second corollary is been done? Because it seems not true to me. For example, Take $G=Z_3$, the cyclic group of three letters. Now consider the representation $ ho : Z_3 \to C^*$ by each element maps to the constant 1. (trivial rep). Now the elements 1 and 2 in $Z_3$ are not conjugate to each other but they are inverse to each other. So despite being members of different conjugacy classes, they produce the same characteristic values which is 1. Let me know if I am wrong. Thanks!
(At time 0:03 ) Could you please explain to me how the second corollary is been done? Because it seems not true to me. For example, Take $G=Z_3$, the cyclic group of three letters. Now consider the representation $
ho : Z_3 \to C^*$ by each element maps to the constant 1. (trivial rep). Now the elements 1 and 2 in $Z_3$ are not conjugate to each other but they are inverse to each other. So despite being members of different conjugacy classes, they produce the same characteristic values which is 1. Let me know if I am wrong. Thanks!
Note that it needs to hold for all characters, and there are two other characters with non-real image (third roots of unity).