This book is based on the new approach to the representation theory of groups developed by J. Q. Chen, F. Wang and M. J. Gao. Chapter 3 covers the central ideas for finite groups (which will be taken over for compact Lie groups later on in chapter 5) and on p. 114-116 you find the basic 7 theorems: Like in quantum mechanics and in the treatment of SO(3) the main objects are the eigenfunctions of complete systems of commuting operators (instead of characters). The main emphasis of this book lies on applications (to nuclear spectroscopy, molecular and crystal groups), so that the permutation group, the rotation group, the unitary groups, point groups and space groups are treated intensively to prepare the reader for explicit calculations. Consequently, most theorems are quoted without proofs (mostly without giving a reference either).
I had some problems with the notation, though. When defining the intrinsic group on p. 73, instead of saying \(\bar R(S)=SR\), the parentheses are missing (so that it looks like a product on both sides) and we are therefore told that this is not an identity but a defining equation, so that e.g. multiplication from the right by another vector T is not allowed! Of course, in Chen’s notation there is no difference between the vectors \(\bar R(ST)\) and \(\bar R(S)T\) which constantly leads to a completely unnecessary confusion. The same holds for the ambiguity with respect to rows and columns on p. 205, where the explanation \(A^{\mu}_{\nu}=A_{\mu \nu}\), \(B^{\nu}_{\mu}=B_{\mu \nu}\) does not help (especially, since these equations do not even hold in general).