Symbol-free definition. The derived subgroup or commutator subgroup of a group is defined in the following equivalent ways: 1  It is the subgroup generated by all commutators of the whole group. 2  It is the normal closure of the subgroup generated by all commutators of the whole group.he notion of derived subgroup or commutator subgroup naturally arose in the context of finding a natural choice for a good composition series for a solvable group -- solvable groups arise very naturally in the fundamental problems dealt with in Galois theory.