  In this paper we explore the possibility of defining p-local finite groups in
terms of transfer properties of their classifying spaces. More precisely, we
consider the question posed by Haynes Miller, whether an equivalent theory can
be obtained by studying triples (f,t,X), where X is a p-complete, nilpotent
space with finite fundamental group, f is a map from the classifying space of a
finite p-group, and t is a stable retraction of f satisfying Frobenius
reciprocity at the level of stable homotopy. We refer to t as a retractive
transfer of f and to (f,t,X) as a retractive transfer triple over S.
  In the case where S is elementary abelian, we answer this question in the
affirmative by showing that a retractive transfer triple (f,t,X) over S does
indeed induce a p-local finite group over S with X as its classifying space.
  Conversely, we show that for a p-local finite group over any finite p-group
S, the natural inclusion of BS into the classifying space of the p-local finite
group has a retractive transfer t, making the triple consisting of the
inclusion, the retractive transfer and the classifying space of the p-local
finite group a retractive transfer triple over S. This also requires a proof,
obtained jointly with Ran Levi, that the classifying space of a p-local finite
group is a nilpotent space, which is of independent interest.
