  Discretized nonabelian gauge theories living on finite group spaces G are
defined by means of a geometric action \int Tr F \wedge *F. This technique is
extended to obtain discrete versions of the Born-Infeld action. The
discretizations are in 1-1 correspondence with differential calculi on finite
groups.
  A consistency condition for duality invariance of the discretized field
equations is derived for discretized U(1) actions S[F] living on a
4-dimensional abelian G. Discretized electromagnetism satisfies this condition
and therefore admits duality rotations.
  Yang-Mills and Born-Infeld theories are also considered on product spaces M^D
x G, and we find the corresponding field theories on M^D after Kaluza-Klein
reduction on the G discrete internal spaces. We examine in some detail the case
G=Z_N, and discuss the limit N -> \infty.
  A self-contained review on the noncommutative differential geometry of finite
groups is included.
