1.6.42 Fields, Groups, Rings

Condition: let \ (F \ Left (X_ {1}, X_ {2}, X_ {3}, X_ {4} \ Right) = X_ {1} X_ {2} X_ {3} \). Prove that the set of \ [g_ {f} = \ left \ {\ sigma \ in s_ {4} \ mid f \ left (x _ \ sigma (1)}, x _ {\ sigma (2)}, x _ {\ sigma (3)}}, X _ {\ SIGMA (4)} \ RIGHT) = \ RIGHT. \] \ (\ left.f \ left (x_ {1}, x_ {2}, x_ {3}, x_ {4} \ right) \ right \} \) is a group.