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 the isotypy induced by the Glauberman–Dade correspondence between blocks of finite groups
Abstract We show that under some conditions the character correspondence introduced by Dade which partly generalizes the Glauberman correspondence induces isotypy between blocks of finite groups.
On Blocks of Characters of Groups of Finite Order: II.
1. The first part of this investigation appeared in these PROCEEDINGS, June, 1946, p. 182.1 In this note, we shall apply our results to a study of the (generalized) decomposition numbers2 of a group G of finite order g and of the arithmetic in the group ring r of G. Let again p be a fixed rational prime number. Select a full system II of elements lro = 1, 7rI, 7r2. in3, ... of orders 1, p, p2, . .. such that every element of an order pa of G iS conjugate in G to exactly one element 1rt of H. Denote by N, the centralizer of iri in G. A full system 2 of elements of G representing the different classes of conjugate elements can be obtained in the following manner: Let oiP, vii) ... represent the different p-regular classes of conjugate elements of N1. Then 2 consists of the elements ria4), 7rio4y', ... for i = 0, 1, 2, 2.3 If ri, r2, ... p rt are the ordinary irreducible characters of G, and if 1, 'P2, . . . are the modular irreducible characters of N1, then for every pregular element a of N1, we have a formula Ed=ido(f) (1)