The set of all such "x" is denoted "XG" and called the "G-invariants" of "X". When "X" is a "G"-module, "XG" is the zeroth cohomology group of "G" with coefficients in "X", and the higher cohomology groups are the derived functors of the functor of "G"-invariants. Given "g" in "G" and "x" in "X" with , 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": 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, , is given by the intersection of the stabilizers "Gx" for all "x" in "X". If "N" is trivial, the action is said to be faithful (or effective). Let "x" and "y" be two elements in "X", and let "g" be a group element such that . Then the two stabilizer groups "Gx" and "Gy" are related by . Proof: by definition, if and only if . Applying "g"−1 to both sides of this equality yields ; that is, . An opposite inclusion follows similarly by taking and supposing . The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, we can associate a conjugacy class of a subgroup of "G" (that is, the set of all conjugates of the subgroup). Let formula_23 denote the conjugacy class of "H". Then the orbit "O" has type formula_23 if the stabilizer formula_25 of some/any "x" in "O" belongs to formula_23. A maximal orbit type is often called a principal orbit type. Orbits and stabilizers are closely related. For a fixed "x" in "X", consider the map "f":"G" → "X" given by "g" ↦ "g"·"x".