  We study the existence of homomorphisms between Out(F_n) and Out(F_m) for n >
5 and m < n(n-1)/2, and conclude that if m is not equal to n then each such
homomorphism factors through the finite group of order 2. In particular this
provides an answer to a question of Bogopol'skii and Puga. In the course of the
argument linear representations of Out(F_n) in dimension less than n(n+1)/2
over fields of characteristic zero are completely classified. It is shown that
each such representation has to factor through the natural projection from
Out(F_n) to GL(n,Z) coming from the action of Out(F_n) on the abelianisation of
F_n. We obtain similar results about linear representation theory of Out(F_4)
and Out(F_5).
