1. Math 3121 Abstract Algebra I Lecture 15 Sections 34-35

2. HW Section 15 Hand in Nov 25: Pages 151: 4, 6, 8, 14, 35, 36 Don’t hand in: Pages 151-: 1, 3, 5, 7, 9, 13, 15, 39

3. HW Section 16 Don’t hand in Page 159-: 1, 2, 3

4. Section 34: Isomorphism Theorems First Isomorphism Theorem Second Isomorphism Theorem Third Isomorphism Theorem

5. First Isomorphism Theorem Theorem (First Isomorphism Theorem): Let φ: G  G’ be a group homomorphism with kernel K, and let  K : G  G/K be the canonical homomorphism. There is a unique isomorphism μ: G/K  φ[G] such that φ(x) = μ(  K (x)) for each x in G. Proof: Section 14

6. Lemma: If N is a normal subgroup of G and if H is any subgroup of G, then H N = N H is a is subgroup of G. Further, if H is normal in G, then H N is normal in G.

7. Second Isomorphism Theorem Theorem (Second Isomorphism Theorem): Let H be a subgroup of a group G, and let N be a normal subgroup of G. Then (H N)/N ≃ H/(H ∩ N). Proof: Let  N : G  G/N be the canonical isomorphism, and let H be a subgroup of G. Then  N [H] is a subgroup of G/N. We will show that both factor groups are isomorphic to  N [H]. Let α be the restriction of  N to H. We claim that the kernel of α is H ∩ N: α ( x ) = e ⇔ x in H and x in N. Thus Ker[ α] = H ∩ N. By the first isomorphism theorem, H/H ∩ N is isomorphic to  N [H]. Let β be the restriction of  N to H N. The kernel of β is N since N is contained in H N. We claim that the image of β is  N [H]: y= β(h) with h in H ⇔ y= β(h) e ⇔ y = β(h x) for all x in N. Thus β[H N] is  N [H]. By the first isomorphism theorem, H N / N is isomorphic to  N [H].

8. Example Given: G = Z × Z × Z H = Z × Z ×{0} N = {0} × Z × Z Then H N = Z × Z × Z H ∩ N = {0} × Z × {0} Thus H N/N = Z × Z × Z/ {0} × Z × Z ≃ Z H/H ∩ N = Z × Z ×{0}/ {0} × Z × {0} ≃ Z

9. Third Isomorphism Theorem Theorem (Third Isomorphism Theorem): Let H and K be normal subgroups of a group G, and let K is a subgroup of H. Then G/H ≃ (G/K)/(H/K). Proof: Let φ: G  (G/K)/(H/K) be defined by φ(x) = (x K)/(H/K), for x in G. φ(x) is onto. It is a homomorphism: φ(x y) = ((x y) K)/(H/K) = ((x K) (y K))/(H/K) = ((x K) ))/(H/K))((y K))/(H/K) ) = φ(x) φ(y) The kernel of φ is H. Thus G/H ≃ (G/K)/(H/K).

10. Example Given K = 6Z H = 2Z G = Z Then G/H = Z/2Z ≃ Z 2 G/K = Z/6Z ≃ Z 6 H/K = 2Z/6Z ≃ Z 3 = {0, 2, 4} in Z 6 (G/K)/(H/K)

11. Example Given G = Z H = n Z K = m n Z Then G/H = Z/n Z ≃ Z n G/K = Z/(n m Z) ≃ Z n m H/K = n Z/(n m) Z ≃ Z m = {0, n, 2n, 3n, …} in Z n m (G/K)/(H/K) ≃ Z n

12. HW for Section 34 Do Hand in (Due Dec 2): Pages 310-311: 2, 4, 7 Don’t hand in: Pages 310-311: 1, 3

13. Section 36: Series of Groups Subnormal and normal series Refinements of series Isomorphic series The Schreier theorem Zassenhaus lemma (butterfly) The Jordan-Holder Theorem

14. Subnormal and Normal Series Definition: A subnormal series of a group G is a finite sequence H 0, H 1, …, H n of subgroups of G such that each H i is a normal subgroup of H i+1. Definition: A normal series of a group G is a finite sequence H 0, H 1, …, H n of normal subgroups of G such that each H i is a subgroup of H i+1.

15. Examples Normal series of Z: {0} < 8 Z < 4 Z < Z {0} < 9 Z < Z Subnormal series of D 4 {ρ 0 } < {ρ 0, μ 1 } < {ρ 0, ρ 2, μ 1, μ 2 } < D 4

16. Refinement Definition: A subnormal (normal) series {K j } is a refinement of a subnormal (normal) series {H i } of a group G if {H i } is a subset of {K j }.

17. Example Normal series of Z: {0} < 8 Z < 4 Z < Z {0} < 9 Z < Z Have refinements {0} < 72 Z < 8 Z < 4 Z < Z {0} < 72 Z < 9 Z < Z

18. Isomorphic Series Definition: Two series {K j } and {H i } of a group G are isomorphic is there is a one-to-one correspondence between {K j+1 /K j } and {H i+1 /H i } such that corresponding factor groups are isomorphic.

19. Example Normal series of Z: {0} < 8 Z < 4 Z < Z {0} < 9 Z < Z Have refinements {0} < 72 Z < 8 Z < 4 Z < Z {0} < 72 Z < 9 Z < Z

20. Butterfly Lemma Lemma (Zassenhaus) Let H and K be subgroups of a group G and let H* and K* be normal subgroups of H and K, respectively. Then 1)H*(H ∩ K*) is a normal subgroup of H*(H ∩ K). 2)K*(H* ∩ K) is a normal subgroup of K*(H ∩ K ). 3)(H ∩ K*) (H* ∩ K) is a normal subgroup of H ∩ K. All three factor groups H*(H ∩ K)/H*(H ∩ K*), K*(H ∩ K )/ K*(H* ∩ K), and H ∩ K/ (H ∩ K*) (H* ∩ K) are isomorphic. Proof: See the book. Needs lemma 34.4

21. Picture of the Butterfly H*(H ∩ K) HK (H ∩ K) H* K* H*(H ∩ K*) K*(H* ∩ K) (H* ∩K)(H* ∩ K) H* ∩ K H ∩ K*

22. The Schreier Theorem Theorem: Two subnormal (normal) series of a group G have isomorphic refinements. Proof: in the book. Sketch: Form refinements and use the butterfly lemma. Define H i,j = H i (H i+1 ∩ K j ) refines H i K j,i = K j (H i ∩ K j+1 ) refines K j

23. Composition Series Definition: A subnormal series {H i } of a group G is a composition series if all the factor group H i+1 /H i are simple. Definition: A normal series {H i } of a group G is a principal or chief series if all the factor group H i+1 /H i are simple.

24. The Jordan-Holder Theorem Theorem (Jordan-Holder): Any two composition (principle) series of a group are isomorphic. Proof: Use Schreier since these are maximally refined.

25. HW on Section 35 Don’t hand in: Pages 319-321: 1, 3, 5, 7 Do hand in: Pages 319-321: 2, 4, 6, 8