I was going through following exercise from Isaacs' character theory. I solved the exercise, but my question here is about a note after it. The exercise is stated below:
Q.1 I tried to give proof of Note; I thought the orthogonality of characters can be proved, but how about orthogonality involving conjugacy classes?
To be precise, the author states following orthogonality relations, among which I couldn't prove (2.18) from (2.20), the above Schur relations. Can we obtain it?
Q.2 Among (2.20), (2.14) and (2.18), which relations are exactly the Frobenius' orthogonality relations and which are Schur's orthogonality relations? Frobenius states in terms of characters perhaps, does Schur also give orthogonality in terms of characters? or matrix representations?