  We extend the duality theory for topological groups from the classical theory
for compact Lie groups, via the topological study by J. R. Klein [Kl01] and the
p-complete study for p-compact groups by T. Bauer [Ba04], to a general duality
theory for stably dualizable groups in the E-local stable homotopy category,
for any spectrum E. The principal new examples occur in the K(n)-local
category, where the Eilenberg-Mac Lane spaces G = K(Z/p, q) are stably
dualizable and nontrivial for 0 <= q <= n.
  We show how to associate to each E-locally stably dualizable group G a stably
defined representation sphere S^{adG}, called the dualizing spectrum, which is
dualizable and invertible in the E-local category. Each stably dualizable group
is Atiyah-Poincare self-dual in the E-local category, up to a shift by S^{adG}.
There are dimension-shifting norm- and transfer maps for spectra with G-action,
again with a shift given by S^{adG}. The stably dualizable group G also admits
a kind of framed bordism class [G] in pi_*(L_E S), in degree dim_E(G) =
[S^{adG}] of the Pic_E-graded homotopy groups of the E-localized sphere
spectrum.
