Normal Subgroups Definition Let G be a group

Normal Subgroups • • • Definition: Let (G, ) be a group and let K be a subgroup of G. Then K is said to be a normal subgroup if g. K = Kg for every g G. Remark: Note that the definition requires g. K = Kg as sets; this is not the same as saying that g k = k g for every k K and for every g G. Alternative versions of the definition: a) g. Kg – 1 = {gkg – 1: k K} = K for every g G or b) g – 1 Kg = K for every g G. In practice, it suffices to show g. Kg – 1 K for every g G. = Notation: K G

Examples of Normal Subgroups 1. {e} and G are normal subgroups for every group G. 2. In an abelian group G, every subgroup H is normal since here g h = h g for every h H and for every g G. 3. If K G and [G: K] = 2, then K is a normal subgroup. In particular, the subgroup <(123)> is a normal subgroup of S 3. 4. However, not all subgroups are normal, for example, <(12)> is not a normal subgroup of S 3.

Homomorphisms Definition: Let (G, ) and (H, ) be groups, and let : G H be a function such that (a b ) = (a) (b) for all a , b G. Then is said to be an homomorphism from G to K. • Remark: this is a generalization of isomorphism, i. e. a composition preserving map (not necessarily bijective). • Definition: Let (G, ) and (H, ) be groups, and let : G H be a homomorphism. Then: i. the image or range of is defined as Im ( ) = {h H: h = (g) for some g G} ii. the kernel of is defined as ker ( ) = {g G : (g) = e’ H, where e’ is the identity element of H} •

Essential Properties of Homorphisms Proposition 14: Let (G, ) and (H, ) be groups, and let : G H be a homomorphism. Then: i. (e) = e’ and (x – 1) = (x) – 1 for all x G. ii. Im ( ) is a subgroup of H. iii. Ker ( ) is a normal subgroup of G. iv. is injective if and only if Ker ( ) = {e} v. Furthermore, if Im ( ) is a finite group, then [G: Ker ( )] = | Im ( ) |. • Proof of i. , iii. is left as an exercise. • Example: The mapping : (Z, +) (Zn , ) given by (a) = a (mod n) = [a]n for all integers a Z is a homomorphism. Im ( ) = Zn, and Ker ( ) = n. Z. •