  As shown in our previous papers (hep-th/0209001 and reference therein),
quantum theory based on a Galois field (GFQT) possesses a new symmetry between
particles and antiparticles, which has no analog in the standard approach. In
the present paper it is shown that this symmetry (called the AB one) is also
compatible with supersymmetry. We believe this is a strong argument in favor of
our assumption that the AB symmetry is a fundamental symmetry in the GFQT (and
in nature if it is described by quantum theory over a Galois field). We also
consider operatorial formulations of space inversion and X inversion in the
GFQT. It is shown in particular that the well known fact, that the parity of
bosons is real and the parity of fermions is imaginary, is a simple consequence
of the AB symmetry.
