Homomorphisms and Normal Subgroups
In Chapter 2 we defined a group on the basis of properties we observed in the examples in Chapter 1. The essentials were a set of elements and an operation with some rather limited properties. We also observed that Examples 1.1 and 1.2 were essentially the same, in the sense that the operations had the same table except for the choice of symbols used to represent the elements. The question, then, is how a set with a group operation might be transformed while preserving some or all of the structure. You should recall that in linear algebra you had linear transformations that preserved the structure of a vector space; functions analogous to linear transformations are basic to group theory. We’ll use the notation ϕ: G → G* to mean that ϕ is a function from G into G*.