A Remark on Orthogonality Relations in Finite Groups
On the basis of Prof. R. Brauer’s fundamental work, certain orthogonality relations for characters of finite groups have recently been studied by Brauer himself, M. Osima, and one of the present writers; see Iizuka [7] and the references there. In the present short note some general remarks on orthogonality relations, dealing with “blocks” and “sections” of general type, are given first. They are of elementary, and often formal, nature and their proofs are merely combinations of known arguments. So, no deep significance is claimed on them, in comparison with the above alluded results based on deeper arithmetico-group-theoretical considerations. However, applied to blocks and sections of such deeper nature, our remarks give some rather useful informations on them. Thus, for instance, the “maximality” feature of 77-blocks is given a formulation (Prop. 5 below) finer than the one given in [71 Further, some new types of blocks and sections can be constructed, again in application of our remarks to such classical ones. These new blocks and sections give thus new orthogonality relations and we hope that some of them may turn to have some significance. There arize also several problems, which are stated at the end of the present note and to some of which we wish to come back elsewhere.
On Brauer's theorem on sections in the theory of blocks of group characters
R. BRAUER, in his earlier p a p ~ [3], announced an important theorem concerning both "blocks" o f group characters and "sections" in groups. In his recent paper [5], he gave a proof of the theorem, in which certain idem.patents of the centers of group rings are used together with linear functions defined on the centers. In this note, we shall give another proof of the theorem by making use of the idempotents (w and, .as a consequence of this, have some results on the idempotents (w
On the Connection Between the Ordinary and The Modular Characters of Groups of Finite Order
The representations of groups by matrices with coefficients in a modular field and the corresponding modular group characters have been studied in two earlier papers;' the aim of the present paper is to continue this work. Let 6 be a group of finite order g and let p be a rational prime. If tj(G), Rz(G), * are the (ordinary) characters of 65, and spo(G), jp2(G), * the modular characters of 6 for p, then we have formulae