  The Yamabe invariant is an invariant of a closed smooth manifold defined
using conformal geometry and the scalar curvature. Recently, Petean showed that
the Yamabe invariant is non-negative for all closed simply connected manifolds
of dimension $\ge 5$. We extend this to show that Yamabe invariant is
non-negative for all closed manifolds of dimension $\ge 5$ with fundamental
group of odd order having all Sylow subgroups abelian. The main new geometric
input is a way of studying the Yamabe invariant on Toda brackets. A similar
method of proof shows that all closed manifolds of dimension $\ge 5$ with
fundamental group of odd order having all Sylow subgroups elementary abelian,
with non-spin universal cover, admit metrics of positive scalar curvature, once
one restricts to the ``complement'' of manifolds whose homology classes are
``toral.'' The exceptional toral homology classes only exist in dimensions not
exceeding the ``rank'' of the fundamental group, so this proves important cases
of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently
large.
