1. (1) Japanese Journal of Mathematics, 17 (1941). Cf. the remarks in the footnote 28) below I want to express my sincere thanks to Messrs. M. Abe and K. Kondo who kindly pointed out those remarks. See also ERRATA at the end.
2. (2) R. Brauer-C. Nesbitt (4), R. Brauer (2) (3); see the references at the end of Part I.
3. (3) R. Brauer (3).
4. (4) I mean a theory detailed and complete to the same degree as that in R. Brauer (3).
5. (5) When we speak simply of irreducible characters or representations we shall always mean ordinary ones.