Investigation on Groups of Even Order, I.

1. Let G be a group of finite order g and let U be a subgroup of order g. If f is a complex-valued class function defined on G, i.e. a function whose value remains constant on each of the conjugate classes K1, K2, . . . K, of G, then f can be expressed by the values of the irreducible characters xi, X2, . . *, Xk of G. If we can find a class function f on G, a class function f on G, and an element a of a such that f(a) = J(a), this yields a relation between the values of xi, X2,..., and the values of the irreducible characters xl, X22 . . . of G for a. 2. We shall assume that the order g is even. Then G contains involutions, i.e. elements of order 2. Let Ka and Kp be two fixed conjugate classes of involutions. For any a EE G, let fp(a) denote the number of ordered pairs (t, ) of elements of G such that