Given g in G and x in X with g ⋅ x = x, it is said that " x is a fixed point of g " or that " g fixes x ". For every x in X, the stabilizer subgroup of G with respect to x (also called the isotropy group or little group) is the set of all elements in G that fix x : G x = { g ∈ ∈ G : g ⋅ ⋅ x = x } . This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial. The kernel N of the homomorphism with the symmetric group, G → Sym(X), is given by the intersection of the stabilizers G x for all x in X. If N is trivial, the action is said to be faithful (or effective).