0. Introduction. In recent years, there has been a general interest in the problem of understanding the homotopy type of function spaces Map (X, Y) where X is an infinite complex. The special case where X is the classifying space of a finite group gives the celebrated Segal's conjecture [4] when Y = lim fQnn = QS' and Sullivan's conjecture [10] when Y is a finite complex. One would like to know whether the affirmative solution to these conjectures can also be applied to the study of Map (X, Y) for other infinite complexes X. The philosophy behind this approach to the problem has already manifested itself in [5], [6], [7] and [8] among others to give interesting results. In this note, we will utilize the solution to Segal's conjecture to compute stable maps from Eilenberg-MacLane spaces to classifying spaces. As an application, we will also describe the homotopy type of Map (X, Y) at a prime p where X = K(1T, n), wT finite, n > 2 and Y = Q(G), G = 0, U, Sp. The original motivation behind this paper was due to Anderson & Hodgkin [1]. There they proved vanishing results on the topological K-theory of higher Eilenberg-MacLane spaces using Atyiah's theorem on the K-theory of the classifying spaces of compact Lie groups. We shall prove that the same result holds for stable cohomotopy groups. The author is especially grateful to Professor Gunnar Carlsson for pointing out [1] and how Segal's conjecture comes into play.