《Discrete Mathematics II教学-华南理工》Lecture 12 Homomorphisms.pdfVIP

III.13 Homomorphisms 1 Part III. Homomorphisms and Factor Groups Section III.13. Homomorphisms Note. So far our study of algebra has been a study of the structure of groups. By “structure” I mean such properties as abelian or nonabelian, the number of generators, the orders of subgroups, the types of subgroups, etc. The idea behind a homomorphism between two groups is that it is a mapping which preserves the binary operation (from which all “structure” follows), but may not be a one to one and onto mapping (and so it may lack the preservation of the “purely set theoretic” properties, as the text says). Definition 13.1. A map φ of a group G into a group G is a homomorphism if for all a, b ∈ G we have φ(ab) = φ(a)φ(b). Note. We see that φ : G → G is an isomorphism if it is a one to one and onto homomorphism. Note. There is always a homomorphism between any two groups G and G . If e is the identity element of G , then φ(g) = e for all g ∈ G is a homomorphism called the trivial homomorphism. III.13 Homomorphisms 2 Example 13.2. Suppose φ : G → G is a homomorphism and φ is onto G . If G is abelian then G is abelian. Notice that this shows how we can get structure preservation without necessarily having an isomorphism. Proof. Let a , b ∈ G . Since φ is onto G , there are a, b ∈ G such that φ(a) = a and φ(b) = b . Now a b = φ(a)φ(b) = φ(ab) s