  The past decade has seen a remarkable resurgence of the old programme of
finding more or less a priori axioms for the mathematical framework of quantum
mechanics. The new impetus comes largely from quantum information theory; in
contrast to work in the older tradition, which tended to concentrate on
structural features of individual quantum systems, the newer work is marked by
an emphasis on systems in interaction. Within this newer work, one can discerne
two distinct approaches: one is "top-down", and attempts to capture in
category-theoretic terms what is distinctive about quantum information
processing. The other is "bottom up", attempting to construct non-classical
models and theories by hand, as it were, and then characterizing those features
that mark out quantum-like behavior. This paper blends these approaches. We
present a constructive, bottom-up recipe for building probabilistic theories
having strong symmetry properties, using as data any uniform enlargement of the
symmetric group $S(E)$ of any set, to a larger group $G(E)$. Subject to some
natural conditions, our construction leads to a monoidal category of fully
symmetric test spaces, in which the monoidal product is "non-signaling".
