Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď X which all map to the same element y P Y.

Transcription

1 Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď X which all map to the same element y P Y.

2 Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď X which all map to the same element y P Y. Example 1: The fibers of the homomorphism are ϕ : Z Ñ Z{4Z z ÞÑ z. 4Z t4z z P Zu t..., 8, 4, 0, 4, 8,... u ÞÑ 0 4Z ` 1 t4z ` 1 z P Zu t..., 7, 3, 1, 5, 9,... u ÞÑ 1 4Z ` 2 t4z ` 2 z P Zu t..., 6, 2, 2, 6, 10,... u ÞÑ 2 4Z ` 3 t4z ` 3 z P Zu t..., 5, 1, 3, 7, 11,... u ÞÑ 3

3 Last time: Example 2: The map ϕ : D 12 Ñ S 3 defined by s ÞÑ p12q r ÞÑ p123q extends to a homomorphism. Its fibers are K K kerpϕq t1, r 3 u ÞÑ 1 Ks tks k P Ku ts, r 3 su ÞÑ p12q Krs tkrs k P Ku trs, r 4 su ÞÑ p13q Kr 2 s tkr 2 s k P Ku tr 2 s, r 5 su ÞÑ p23q Kr tkr k P Ku tr, r 4 u ÞÑ p123q Kr 2 tkr 2 k P Ku tr 2, r 5 u ÞÑ p132q

4 Last time: Let K ď G. Then for g P G, we call the sets gk tgk k P Ku and Kg tkg k P Ku the left and right coset of K (corresponding to g). And element of a coset is called a representative of the coset. Theorem Let ϕ : G Ñ H be a surjective homomorphism of groups with kernel K. For each h P H, let 1. Then X h ϕ 1 phq tg P G ϕpgq hu. x P X a and y P X b implies xy P X ab. In particular as subsets of G, tx h h P Hu is a group under the operation X a X b X ab. (We call this group the quotient group G{kerpϕq). 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx.

5 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx. This says that if K ď G is the kernel of some homomorphism, then xk Kx for all x P G, and Kx Ky for all y P Kx (the sets are equal).

6 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx. This says that if K ď G is the kernel of some homomorphism, then xk Kx for all x P G, and Kx Ky for all y P Kx (the sets are equal). Caution: Kernels are special left and right cosets, even of subgroups, aren t always equal.

7 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx. This says that if K ď G is the kernel of some homomorphism, then xk Kx for all x P G, and Kx Ky for all y P Kx (the sets are equal). Caution: Kernels are special left and right cosets, even of subgroups, aren t always equal. For example, consider G S 3, H xp12qy t1, p12qu, and x p23q.

8 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx. This says that if K ď G is the kernel of some homomorphism, then xk Kx for all x P G, and Kx Ky for all y P Kx (the sets are equal). Caution: Kernels are special left and right cosets, even of subgroups, aren t always equal. For example, consider These give G S 3, H xp12qy t1, p12qu, and x p23q. xh tp23q1, p23qp12qu

9 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx. This says that if K ď G is the kernel of some homomorphism, then xk Kx for all x P G, and Kx Ky for all y P Kx (the sets are equal). Caution: Kernels are special left and right cosets, even of subgroups, aren t always equal. For example, consider These give G S 3, H xp12qy t1, p12qu, and x p23q. xh tp23q1, p23qp12qu tp23q, p132qu

10 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx. This says that if K ď G is the kernel of some homomorphism, then xk Kx for all x P G, and Kx Ky for all y P Kx (the sets are equal). Caution: Kernels are special left and right cosets, even of subgroups, aren t always equal. For example, consider These give and G S 3, H xp12qy t1, p12qu, and x p23q. xh tp23q1, p23qp12qu tp23q, p132qu Hx t1p23q, p12qp23qu

11 2. Fix some fiber X h. For any x P X h, X h xk and X h Kx. This says that if K ď G is the kernel of some homomorphism, then xk Kx for all x P G, and Kx Ky for all y P Kx (the sets are equal). Caution: Kernels are special left and right cosets, even of subgroups, aren t always equal. For example, consider These give and G S 3, H xp12qy t1, p12qu, and x p23q. xh tp23q1, p23qp12qu tp23q, p132qu Hx t1p23q, p12qp23qu tp23q, p123qu.

12 1. Then x P X a and y P X b implies xy P X ab. In particular as subsets of G, tx h h P Hu is a group under the operation X a X b X ab. This says that if K is the kernel of some homomorphism of a group G, then the set G{K : txk x P Gu is a group under the multiplication xk yk pxyqk.

13 1. Then x P X a and y P X b implies xy P X ab. In particular as subsets of G, tx h h P Hu is a group under the operation X a X b X ab. This says that if K is the kernel of some homomorphism of a group G, then the set G{K : txk x P Gu is a group under the multiplication xk yk pxyqk. Further, if K is, specifically, the kernel of the homomorphism ϕ : G Ñ H, then the map G{K Ñ imgpϕq ď H defined by xk ÞÑ ϕpxq is a bijective homomorphism

14 1. Then x P X a and y P X b implies xy P X ab. In particular as subsets of G, tx h h P Hu is a group under the operation X a X b X ab. This says that if K is the kernel of some homomorphism of a group G, then the set G{K : txk x P Gu is a group under the multiplication xk yk pxyqk. Further, if K is, specifically, the kernel of the homomorphism ϕ : G Ñ H, then the map G{K Ñ imgpϕq ď H defined by xk ÞÑ ϕpxq is a bijective homomorphism, so G{kerpϕq imgpϕq. (This is called the 1st isomorphism theorem)

15 Skip the homomorphism When do the set of cosets of a set A Ď G form a group? i.e. when is the multiplication well-defined? xa ya pxyqa ( )

16 Skip the homomorphism When do the set of cosets of a set A Ď G form a group? i.e. when is the multiplication well-defined? xa ya pxyqa ( ) No hope if A is not a group!

17 Skip the homomorphism When do the set of cosets of a set A Ď G form a group? i.e. when is the multiplication well-defined? Proposition Let H ď G. xa ya pxyqa ( ) No hope if A is not a group! 1. The left cosets of H partition the elements of G. 2. xh yh if and only if y 1 x P H.

18 Skip the homomorphism When do the set of cosets of a set A Ď G form a group? i.e. when is the multiplication xa ya pxyqa ( ) well-defined? Proposition Let H ď G. No hope if A is not a group! 1. The left cosets of H partition the elements of G. 2. xh yh if and only if y 1 x P H. For 1, consider the equivalence relation x y if x P yh...

19 Skip the homomorphism When do the set of cosets of a set A Ď G form a group? i.e. when is the multiplication xa ya pxyqa ( ) well-defined? Proposition Let H ď G. No hope if A is not a group! 1. The left cosets of H partition the elements of G. 2. xh yh if and only if y 1 x P H. For 1, consider the equivalence relation x y if x P yh... Proposition Let H ď G. 1. The operation in ( ) is well defined if and only if gxg 1 P H for all x P H and g P G. 2. If is well-defined, then G{H tgh g P Gu forms a group with 1 1H and pghq 1 g 1 H.

20 Definition A subgroup N ď G is normal if gng 1 N for all g P G, i.e. when G{N is a well-defined group. Write N G.

21 Definition A subgroup N ď G is normal if gng 1 N for all g P G, i.e. when G{N is a well-defined group. Write N G. In summary, let N ď G. Then the following are equivalent: 1. N is normal in G 2. N G pnq G 3. gn Ng for all g P G 4. the operation on left cosets given by xn yn pxyqn is well-defined 5. gng 1 Ď N for all g P G

22 Example: Letting D 8 act on itself by conjugation (g a gag 1 ) yields the following table: Ð h Ñ ghg 1 1 r r 2 r 3 s sr sr 2 sr r r 2 r 3 s sr sr 2 sr 3 r 1 r r 2 r 3 sr 2 sr 3 s sr Ò r 2 1 r r 2 r 3 s sr sr 2 sr 3 g r 3 1 r r 2 r 3 sr 2 sr 3 s sr Ó s 1 r 3 r 2 r s sr 3 sr 2 sr sr 1 r 3 r 2 r sr 2 sr s sr 3 sr 2 1 r 3 r 2 r s sr 3 sr 2 sr sr 3 1 r 3 r 2 r sr 2 sr s sr 3 What are the normal subgroups of D 8?

23 Example: Letting D 8 act on itself by conjugation (g a gag 1 ) yields the following table: Ð h Ñ Ò g Ó ghg 1 1 r r 2 r 3 s sr sr 2 sr r r 2 r 3 s sr sr 2 sr 3 r 1 r r 2 r 3 sr 2 sr 3 s sr r 2 1 r r 2 r 3 s sr sr 2 sr 3 r 3 1 r r 2 r 3 sr 2 sr 3 s sr s 1 r 3 r 2 r s sr 3 sr 2 sr sr 1 r 3 r 2 r sr 2 sr s sr 3 sr 2 1 r 3 r 2 r s sr 3 sr 2 sr sr 3 1 r 3 r 2 r sr 2 sr s sr 3 What are the normal subgroups of D 8? Note: subgroup is transitive but normal is not!! For example, t1, su IJ t1, r 2, s, sr 2 u and t1, r 2, s, sr 2 u IJ D 8, but t1, su đ D 8.

24 More examples 1. G and 1 t1 G u are always normal in G, with G{1 G and G{G 1.

25 More examples 1. G and 1 t1 G u are always normal in G, with G{1 G and G{G All subgroups of the center ZpGq of a group are normal in G.

26 More examples 1. G and 1 t1 G u are always normal in G, with G{1 G and G{G All subgroups of the center ZpGq of a group are normal in G. 3. All subgroups of abelian groups are normal.

27 More examples 1. G and 1 t1 G u are always normal in G, with G{1 G and G{G All subgroups of the center ZpGq of a group are normal in G. 3. All subgroups of abelian groups are normal. 4. Quotients of abelian groups are abelian and quotients of cyclic groups are cyclic.

28 You try: Recall Q 8 is generated by 1, i, j, and k, with relations p 1q 2 1, i 2 j 2 k 2 1, i j k j i, j k i k j, k i j i k. Let H x 1y. (1) Show that H IJ Q 8, and conclude Q 8 {H is a group. (2) List the elements of Q 8 {H (pick one rep. per coset). (3) Give a multiplication table for Q 8 {H, and give a presentation for Q 8 {H. (4) Show Q 8 {H Z 2 ˆ Z 2 (where Z 2 xx x 2 1y).

29 You try: Recall Q 8 is generated by 1, i, j, and k, with relations p 1q 2 1, i 2 j 2 k 2 1, i j k j i, j k i k j, k i j i k. Let H x 1y. (1) Show that H IJ Q 8, and conclude Q 8 {H is a group. (2) List the elements of Q 8 {H (pick one rep. per coset). (3) Give a multiplication table for Q 8 {H, and give a presentation for Q 8 {H. (4) Show Q 8 {H Z 2 ˆ Z 2 (where Z 2 xx x 2 1y). H ih jh kh H H ih jh kh ih ih H kh jh jh jh kh H ih kh kh jh ih H

30 You try: Recall Q 8 is generated by 1, i, j, and k, with relations p 1q 2 1, i 2 j 2 k 2 1, i j k j i, j k i k j, k i j i k. Let H x 1y. (1) Show that H IJ Q 8, and conclude Q 8 {H is a group. (2) List the elements of Q 8 {H (pick one rep. per coset). (3) Give a multiplication table for Q 8 {H, and give a presentation for Q 8 {H. (4) Show Q 8 {H Z 2 ˆ Z 2 (where Z 2 xx x 2 1y). H ih jh kh H H ih jh kh ih ih H kh jh jh jh kh H ih kh kh jh ih H Q 8 {H ÝÑ Z 2 ˆ Z 2 H ÞÑ p1, 1q ih ÞÑ px, 1q jh ÞÑ p1, xq kh ÞÑ px, xq

31 You try: Recall Q 8 is generated by 1, i, j, and k, with relations p 1q 2 1, i 2 j 2 k 2 1, i j k j i, j k i k j, k i j i k. Let H x 1y. (1) Show that H IJ Q 8, and conclude Q 8 {H is a group. (2) List the elements of Q 8 {H (pick one rep. per coset). (3) Give a multiplication table for Q 8 {H, and give a presentation for Q 8 {H. (4) Show Q 8 {H Z 2 ˆ Z 2 (where Z 2 xx x 2 1y). H ih jh kh H H ih jh kh ih ih H kh jh jh jh kh H ih kh kh jh ih H Notation: For N IJ G, write x xn for short. Q 8 {H ÝÑ Z 2 ˆ Z 2 H ÞÑ p1, 1q ih ÞÑ px, 1q jh ÞÑ p1, xq kh ÞÑ px, xq

32 You try: Recall Q 8 is generated by 1, i, j, and k, with relations p 1q 2 1, i 2 j 2 k 2 1, i j k j i, j k i k j, k i j i k. Let H x 1y. (1) Show that H IJ Q 8, and conclude Q 8 {H is a group. (2) List the elements of Q 8 {H (pick one rep. per coset). (3) Give a multiplication table for Q 8 {H, and give a presentation for Q 8 {H. (4) Show Q 8 {H Z 2 ˆ Z 2 (where Z 2 xx x 2 1y). 1 ī j k Q 8 {H ÝÑ Z 2 ˆ Z ī j k 1 ÞÑ p1, 1q ī ī 1 k j ī ÞÑ px, 1q j j k 1 ī j ÞÑ p1, xq k k j ī 1 k ÞÑ px, xq Notation: For N IJ G, write x xn for short.