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 0-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 0-functions may be regarded as group-characters of a sort. Explicitly, if x is any p-element of G and VI any irreducible Brauer character of the centralizer C(x) of x in G, there is a unique function 0 defined on G such that: (a) 0 is a class-function on G; (b) 4(xy) = +f(y) if y is p-regular in C(x); (c) O(x1y) = 0 if x1 is a p-element of G which is not conjugate to x and if y is p-regular in C(x1). We call these functions 0 the 0-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 il 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 1D-functions of G, starting from principal indecomposable characters P 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, , Xk are the (ordinary) irreducible characters of G, there are unique complex numbers dij such that