We shall now investigate homomorphic mappings of one group onto another with special attention to the action of the mapping on the subgroups of a group. Certain subgroups have played an important role in the development and application of group theory. Galois, in 1830, discovered the significance of these special groups, the so-called normal (or self-conjugate, or invariant) subgroups, in the course of his investigation of the nature of the roots of algebraic equations. Galois showed that to each algebraic equation there corresponds a group of finite order, and the nature of the roots of the equation depends on the character of the normal subgroups of the group of the equation; that is, the normal subgroups provide the basis for determining the character of the solutions of the associated algebraic equation.

We shall now examine normal subgroups from two points of view: (1) homomorphic mapping, and (2) decomposition of a group into cosets with respect to a normal subgroup. Both approaches will be seen to correspond to different aspects of the same fundamental structural property. The use of approach (1) relies on the working out of detailed relations among group elements by “computing” in accordance with the group axioms. We have already done such computing; for example, solving group equations, and arriving at defining relations of a group.