  We construct a $2\times 2$ matrix algebra as representation of functions on
discrete group $Z_2$ and develop the gauge theory on discrete group proposed by
Starz in the matrix algebra. Accordingly, we show that the non-commutative
geometry model built by R.Conquereax, G.Esposito-Farese and G.Vaillant results
from this approach directly.
  For the purpose of Physical model building, we introduce a free fermion
  Lagrangian on $M_4\times Z_2$ and study Yang-Mills like gauge theory.
