《Discrete Mathematics II教学-华南理工》Lecture 3 Groups II.pptVIP

Lemma 14 Every subgroup of an Abelian group is a normal subgroup. Proof : Let N be a subgroup of an Abelian group G. Then xnx-1 = (xn)x-1 = (nx)x-1 = n(xx-1) = ne = n for all n?N and x?G, where e is the identity element of G. The result follows. Quotient group Definition: Let H be a normal subgroup of a group G. We define the quotient group G/H as follows: The elements of G/H are the cosets of H in G (left or right doesn’t matter, since H is normal); The group operation is defined by (Hx)(Hy) = Hxy for all x,y ?G; in other words, to multiply cosets, we multiply their representatives. Homomorphism Two groups are related in the strongest possible way if they are isomorphic; however, a weaker relationship may exist between two groups. If we relax the requirement that an isomorphism of groups be bijective, we have a homomorphism. Definition: A homomorphism between groups (G,* ) and (H,o ) is a map ? : G?H such that ?(g1* g2) = ?(g1) o?(g2) for g1, g2 ?G. The range of ? in H is called the homomorphic image of ?. Example Let G be a group and g?G. Define a map ? : Z?G by ?(n) = gn. Then ? is a group homomorphism, since ?(m + n) = gm+n = gmgn = ?(m) ?(n). This homomorphism maps Z onto the cyclic subgroup of G generated by g. Properties of group homomorphisms Let ? : G1?G2 be a homomorphism of groups. Then 1. If e is the identity of G1, then ?(e) is the identity of G2; 2. For any element g?G1, ?(g-1) = [?(g)] -1; 3. If H1 is a subgroup of G1, then ?(H1) is a subgroup of G2; 4. If H2 is a subgroup of G2, then ?-1 (H2) = {g?G1 : ?(g)? H2} is a subgroup of G1. Furthermore, if H2 is normal in G2, then ?-1 (H2) is normal in G1. Properties of group homomorphisms(Proof) Proof. (1) Suppose that e and e’ are the identities of G1 and G2, respectively; then e’?(e) = ?(e) = ?(ee) = ?(e) ?(e). By cancellation, ?(e) = e’. (2) This statement follows from the fact that ?(g-1) ?(g) = ?(g-1g) = ?(e) = e: (3) The set ?(H1) is nonempty since the identity of H2 is in ?(H1). Suppose that