  This paper is concerned with the topological space of normalized
quaternion-valued positive definite functions on an arbitrary abelian group G,
especially its convex characteristics. There are two main results. Firstly, we
prove that the extreme elements in the family of such functions are exactly the
homomorphisms from G to the sphere group S, i.e., the unit 3-sphere in the
quaternion algebra. Secondly, we real a phenomenon which does not exist in the
complex setting: The compact convex set of such functions is not a Bauer
simplex except when G is of exponent less than or equal to 2. In contrast, its
complex counterpart is always a Bauer simplex, as is well known. We also
present an integral representation for such functions as an application and
some other minor interesting results.
