  Similar to the three-qubit Greenberger-Horne-Zeilinger (GHZ) symmetry we
explore the four-qubit GHZ symmetry group and its subgroup called restricted
GHZ symmetry group. While the set of symmetric states under the whole group
transformation is represented by three real parameters, the set of symmetric
states under the subgroup transformation is represented by two real parameters.
After comparing the symmetric states for whole and subgroup, the entanglement
is examined for the latter set. It is shown that the set has only two SLOCC
classes, $L_{abc_2}$ and $G_{abcd}$. Extension to the multi-qubit system is
briefly discussed.
