  We prove that for an embedded unstable one-sided minimal hypersurface of the
$(n+1)$-dimensional real projective space, the Morse index is at least $n+2$,
and this bound is attained by the cubic isoparametric minimal hypersurfaces. We
also show that there exist closed embedded two-sided minimal surfaces in the
3-dimensional real projective space of each odd index by computing the index of
the Lawson surfaces.
