  The discrete symmetries of the Dirac field on the de Sitter manifold are
studied taking into account that this has two portions that can play the role
of physical space-times, namely the expanding and a collapsing universes. The
proper discrete isometries which preserve the portion have a physical meaning
in contrast to the improper ones which change the portion remaining thus of a
mere mathematical interest. The discrete symmetries generated by the proper
isometries and charge conjugation are studied in physical frames on the
expanding portion shoving that all the discrete transformations reversing the
cosmic time are local depending on a local boost matrix. Thus the discrete
group of Dirac's theory in the de Sitter expanding universe is obtained showing
that this is of the order 16 having a multiplication table similar to that of
Dirac's theory in special relativity. Moreover, all the discrete de Sitter
isometries, including the improper ones, are studied in conformal frames for
obtaining a global image about the de Sitter isometries in spite of the fact
that these cannot be gathered in a larger discrete group with physical meaning.
