Abstract Algebra Theory and Applications - Computer Science ...

156 CHAPTER 9 HOMOMORPHISMS AND FACTOR GROUPSWe use homomorphisms to study relationships such as the one we have justdescribed.Example 6. Let G be a group and g ∈ G. Define a map φ : Z → G byφ(n) = g n . Then φ is a group homomorphism, sinceφ(m + n) = g m+n = g m g n = φ(m)φ(n).This homomorphism maps Z onto the cyclic subgroup of G generated by g.Example 7. Let G = GL 2 (R). If( a bA =c d)is in G, then the determinant is nonzero; that is, det(A) = ad − bc ≠ 0.Also, for any two elements A and B in G, det(AB) = det(A) det(B). Usingthe determinant, we can define a homomorphism φ : GL 2 (R) → R ∗ byA ↦→ det(A). Example 8. Recall that the circle group T consists of all complex numbersz such that |z| = 1. We can define a homomorphism φ from the additivegroup of real numbers R to T by φ : θ ↦→ cos θ + i sin θ. Indeed,φ(α + β) = cos(α + β) + i sin(α + β)= (cos α cos β − sin α sin β) + i(sin α cos β + cos α sin β)= (cos α + i sin α) + (cos β + i sin β)= φ(α)φ(β).Geometrically, we are simply wrapping the real line around the circle in agroup-theoretic fashion.The following proposition lists some basic properties of group homomorphisms.Proposition 9.3 Let φ : G 1 → G 2 be a homomorphism of groups. Then1. If e is the identity of G 1 , then φ(e) is the identity of G 2 ;2. For any element g ∈ G 1 , φ(g −1 ) = [φ(g)] −1 ;3. If H 1 is a subgroup of G 1 , then φ(H 1 ) is a subgroup of G 2 ;