Math 3121 Abstract Algebra I

Math 3121Abstract Algebra I Lecture 15 Sections 34-35

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

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

Section 34: Isomorphism Theorems • First Isomorphism Theorem • Second Isomorphism Theorem • Third Isomorphism Theorem

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

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.

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 αbe the restriction of N to H. We claim that the kernel of αis 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 βbe the restriction of N to H N. The kernel of βis 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].

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

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).

Example • Given K = 6Z H = 2Z G = Z • Then G/H = Z/2Z ≃ Z2 G/K = Z/6Z ≃ Z6 H/K = 2Z/6Z ≃ Z3 = {0, 2, 4} in Z6 (G/K)/(H/K)

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

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

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

Subnormal and Normal Series Definition: A subnormal series of a group G is a finite sequence H0, H1, …, Hn of subgroups of G such that each Hi is a normal subgroup of Hi+1. Definition: A normal series of a group G is a finite sequence H0, H1, …, Hn of normal subgroups of G such that each Hi is a subgroup of Hi+1.

Examples • Normal series of Z: {0} < 8 Z < 4 Z < Z {0} < 9 Z < Z • Subnormal series of D4 {ρ0} < {ρ0, μ1} < {ρ0 , ρ2, μ1 , μ2} < D4

Refinement Definition: A subnormal (normal) series {Kj} is a refinement of a subnormal (normal) series {Hi} of a group G if {Hi} is a subset of {Kj}.

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

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

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

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 • H*(H ∩ K*) is a normal subgroup of H*(H ∩ K). • K*(H* ∩ K) is a normal subgroup of K*(H ∩ K). • (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

Picture of the Butterfly H K H*(H ∩ K) H*(H ∩ K) (H ∩ K) H*(H ∩ K*) K*(H* ∩ K) (H* ∩K)(H* ∩ K) K* H* H ∩ K* H* ∩ K

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 Hi,j = Hi (Hi+1∩ Kj) refines Hi Kj,i = Kj (Hi∩ Kj+1) refines Kj

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

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.

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

Math 3121 Abstract Algebra I