Factor Groups and Homomorphisms
In the previous chapter, we tended to consider just one group at a time. But we need to find ways of relating groups to each other. For instance, we would like to know if two groups are, in every meaningful sense, the same. This would be the case if we took a group and created a new one by simply changing the labels on the group elements, but left the structure otherwise intact. Surely, we would not wish to think of these as different sorts of groups. This is where the notion of a group homomorphism and, in particular, an isomorphism, will come into the picture. But first, we will discuss factor groups. These constitute an important way of creating new groups from old ones. As we shall see, there is a natural connection between factor groups and homomorphisms. In order to define a factor group, we require a special sort of subgroup, called a normal subgroup. Let us begin there.