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.
A GENERALIZATION OF BRAUER CHARACTERSO)
1. Statement of results. The Brauer characters, or modular characters, of a finite group G with respect to a prime p are defined only on the p-regular elements of G. Brauer [1], [3] has overcome this limitation to some extent by using the Brauer characters of certain subgroups of G, in terms of which he has defined the generalized decomposition matrix D of G. In this paper we study some complex-valued class-functions on G which are closely related to these Brauer characters of subgroups. These functions, which we call (/»-functions, were defined by Brauer in [3, (7D)]. They behave in many ways like Brauer characters; for example, they have orthogonality relations (§4) and they can be distributed among the p-blocks of G (§9). In fact, our central motivating idea has been that the (/»-functions may be regarded as group-characters of a sort. Explicitly, if x is any p-element of G and \¡t any irreducible Brauer character of the centralizer C(x) of x in G, there is a unique function (b defined on G such that : (a) 4> is a class-function on G; (b) <p(xy) = \¡i(y) if y is p-regular in C(x) ; (c) <b(xyy) = 0 if Xy is a p-element of G which is not conjugate to x and if y is p-regular in C(xf). We call these functions <b the ^-functions of G for the prime p. (Actually the definition of Brauer characters depends on the choice of a prime ideal divisor p of p in a suitable algebraic number field [6, p. 589] ; accordingly, the (/»-functions also depend on p.) For x = 1, <b is simply the Brauer character \¡i of G, if we extend the latter by giving it the value 0 on all p-singular elements (see the redefinition of Brauer characters in §2). Dually we define the ^-functions of G, starting from principal indecomposable characters *? of C(x) ; for x = 1, these are the principal indecomposable characters of G. G has exactly k distinct (/»-functions, where k is the number of conjugate classes of G. If Xi, •••,& are the (ordinary) irreducible characters of G, there are unique complex numbers dt) such that
The Globalization problem for inner automorphisms and Skolem-Noether theorems
We investigate the circumstances under which an inner automorphism of a ring with local units can be built from “local” information. Specifically, we consider three natural “inner” type properties for an automorphism of a ring with local units. We show that every inner automorphism is locally inner but the converse is false, even if the automorphism is “piecewise” inner. On the other hand, we construct a large class of rings for which every locally inner automorphism is actually inner. Finally we obtain some Skolem-Noether type Theorems for infinite matrix and triangular matrix rings.