A Characterization of the Characters of Groups of Finite Order

Abstract

Let (5 be a group of finite order g. By a generalized character of (M, we shall mean a linear combination of the (ordinary) irreducible characters of (M with integral rational coefficients. The main purpose of this paper is a characterization of the generalized characters. It will be convenient to call a group e an elementary group, if e is the direct product of a cyclic group I of order a and a p-group 3 of order pr where a is not divisible by the prime number p. We shall then prove THEOREM 1. A complex-valued function 0(G) defined on (5 is a generalized character of (5, if and only if the following two conditions are satisfied (I) 0 is a class function, i.e., the value of O(G) is constant for the elements of each class of conjugate elements of (X. (II) For every elementary subgroup e of (D, the restriction of 0 to e is a generalized character of A. The necessity of these conditions is obvious. The sufficiency is equivalent with a theorem on induced characters obtained in an earlier paper.' However, since the proof of this earlier theorem can be simplified considerably, I will prove Theorem 1 here without reference to the previous paper (Sections 2 and 3). As an immediate consequence of Theorem 1, we have THEOREM 2. The function 0 defined on @5 is an irreducible character of (M, if and only if besides conditions (I) and (II) of Theorem 1, the following further conditions are satisfied (III) The average value of I 0 12 is 1;