  Motivated by the studies of the superconducting pairing states in the
iron-based superconductors, we analyze the effects of Brillouin zone folding
procedure from a space group symmetry perspective for a general class of
materials with the $P4/nmm$ space group. The Brillouin zone folding amounts to
working with an effective one-Fe unit cell, instead of the crystallographic
two-Fe unit cell. We show that the folding procedure can be justified by the
validity of a glide reflection symmetry throughout the crystallographic
Brillouin zone and by the existence of a minimal double degeneracy along the
edges of the latter. We also demonstrate how the folding procedure fails when a
local spin-orbit coupling is included although the latter does not break any of
the space group symmetries of the bare Hamiltonian. In light of these general
symmetry considerations, we further discuss the implications of the glide
reflection symmetry for the superconducting pairing in an effective
multi-orbital $t-J_{1}-J_{2}$ model. We find that the $P4/nmm$ space group
symmetry allows only pairing states with even parity under the glide reflection
and zero total momentum.
