Math 3121Abstract Algebra I

Lecture 15Sections 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 inPage 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 TheoremTheorem (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].

Example

• Given:G = Z × Z × ZH = Z × Z ×{0}N = {0} × Z × Z

• ThenH N = Z × Z × ZH∩N = {0} × Z × {0}

• ThusH N/N = Z × Z × Z/ {0} × Z × Z ≃ ZH/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

• GivenK = 6ZH = 2ZG = Z

• ThenG/H = Z/2Z ≃ Z2

G/K = Z/6Z ≃ Z6

H/K = 2Z/6Z ≃ Z3 = {0, 2, 4} in Z6

(G/K)/(H/K)

Example

• GivenG = ZH = n ZK = m n Z

• ThenG/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. Then1) 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

Picture of the Butterfly

H*(H ∩ K) H*(H ∩ K)

H K

(H ∩ K)

H*K*

H*(H ∩ K*) K*(H* ∩ K)

(H* ∩K)(H* ∩ K)

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