《Discrete Mathematics II教学-华南理工》Lecture 13 Factor Groups.pdfVIP

III.14 Factor Groups 1 Section III.14. Factor Groups Note. In the previous section we used a homomorphism φ to map the cosets of H = Ker(φ) one to one and into group G where φ : G → G . In fact, φ is one to one and onto φ[G], which we know to be a group by Theorem 13.2 Part (3). So we should be able to make a group out of the cosets of H = Ker(φ) and the group should be isomorphic to φ[G]. In this section, we make this clear and explicitly define the binary operation on the cosets. The group of cosets of H = Ker(φ) is called a factor group . Theorem 14.1. Let φ : G → G be a group homomorphism with kernel H = Ker(φ). Then the cosets of H = Ker(φ) form a factor group , G/H , where (aH ) · (bH ) = (ab)H . Also, the map µ : G/H → φ[G] defined by µ(aH ) = φ(a) is an isomorphism. Both coset multiplication and µ are well defined (i.e., independent of the choices of a and b from the cosets). Example 14.2. Define γ : Z → Zn as γ (m) ≡ m (mod n). Then γ is a homomor- phism by Example 13.10. Ker(γ) = {m ∈ Z | m ≡ 0(mod n)} = nZ. The cosets of Ker(γ) = nZ are (remember, Z and Zn are both additive groups): nZ = 0 + nZ = {. . . , −2n, −n, 0, n, 2n, . . . } 1 + nZ = {. . . , −2n + 1, −n + 1, 1, n + 1, 2n + 1, . . .} 2 + nZ = {. . . , −2n + 2, −n + 2, 2, n + 2, 2n + 2, . . .} . . . III.14 Factor Groups 2 (n − 1) + nZ = {. . . , −n − 1