  Using a ``3 by 3 matrix trick'' we previously showed that multiplication in a
C*-algebra A, an algebraic structure, is determined by the geometry of the
C*-algebra of the 3 by 3 matrices with entries from A. As an application of
this algebra-geometry duality we now construct an order theoretic based duality
theory for all groups which are either locally compact abelian or finite. This
construction generalizes the van Kampen--Pontriagin duality for locally compact
abelian groups.
