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Math 3121 Abstract 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 - PowerPoint PPT Presentation

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  • Math 3121Abstract Algebra ILecture 15Sections 34-35

  • HW Section 15Hand in Nov 25:Pages 151: 4, 6, 8, 14, 35, 36Dont hand in:Pages 151-: 1, 3, 5, 7, 9, 13, 15, 39

  • HW Section 16Dont hand inPage 159-: 1, 2, 3

  • Section 34: Isomorphism TheoremsFirst Isomorphism TheoremSecond Isomorphism TheoremThird Isomorphism Theorem

  • First Isomorphism TheoremTheorem (First Isomorphism Theorem): Let : G G be a group homomorphism with kernel K, and let K: G G/K be the canonical homomorphism. There is a unique isomorphism : G/K [G] such that (x) = (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 be the restriction of N to H. We claim that the kernel of is H N: (x) = e x in H and x in N. Thus Ker[] = H N. By the first isomorphism theorem, H/H N is isomorphic to N[H].

    Let be the restriction of N to H N. The kernel of is N since N is contained in H N. We claim that the image of is N[H]:y= (h) with h in H y= (h) e y = (h x) for all x in N. Thus [H N] is N[H].By the first isomorphism theorem, H N / N is isomorphic to N[H].

  • ExampleGiven:G = Z Z ZH = Z Z {0}N = {0} Z ZThenH N = Z Z ZHN = {0} Z {0}ThusH N/N = Z Z Z/ {0} Z Z ZH/HN = Z Z {0}/ {0} Z {0} Z

  • Third Isomorphism TheoremTheorem (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 : G (G/K)/(H/K) be defined by (x) = (x K)/(H/K), for x in G. (x) is onto. It is a homomorphism: (x y) = ((x y) K)/(H/K) = ((x K) (y K))/(H/K) = ((x K) ))/(H/K))((y K))/(H/K) )= (x) (y) The kernel of is H. Thus G/H (G/K)/(H/K).

  • ExampleGivenK = 6ZH = 2ZG = ZThenG/H = Z/2Z Z2G/K = Z/6Z Z6H/K = 2Z/6Z Z3 = {0, 2, 4} in Z6 (G/K)/(H/K)

  • ExampleGivenG = ZH = n ZK = m n ZThenG/H = Z/n Z ZnG/K = Z/(n m Z) Zn mH/K = n Z/(n m) Z Zm = {0, n, 2n, 3n, } in Zn m (G/K)/(H/K) Zn

  • HW for Section 34Do Hand in (Due Dec 2):Pages 310-311: 2, 4, 7Dont hand in:Pages 310-311: 1, 3

  • Section 36: Series of GroupsSubnormal and normal seriesRefinements of seriesIsomorphic seriesThe Schreier theoremZassenhaus lemma (butterfly)The Jordan-Holder Theorem

  • Subnormal and Normal SeriesDefinition: 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.

  • ExamplesNormal series of Z:{0} < 8 Z < 4 Z < Z{0} < 9 Z < ZSubnormal series of D4{0} < {0, 1} < {0 , 2, 1 , 2} < D4

  • RefinementDefinition: 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}.

  • ExampleNormal series of Z:{0} < 8 Z < 4 Z < Z{0} < 9 Z < ZHave refinements{0} < 72 Z < 8 Z < 4 Z < Z{0} < 72 Z < 9 Z < Z

  • Isomorphic SeriesDefinition: 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.

  • ExampleNormal series of Z:{0} < 8 Z < 4 Z < Z{0} < 9 Z < ZHave refinements{0} < 72 Z < 8 Z < 4 Z < Z{0} < 72 Z < 9 Z < Z

  • Butterfly LemmaLemma (Zassenhaus) Let H and K be subgroups of a group G and let H* and K* be normal subgroups of H and K, respectively. ThenH*(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 ButterflyH*(H K)H*(H K)HK(H K)H*K*H*(H K*)K*(H* K)(H* K)(H* K)H* KH K*

  • The Schreier TheoremTheorem: 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 HiKj,i = Kj (Hi Kj+1) refines Kj

  • Composition SeriesDefinition: 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 TheoremTheorem (Jordan-Holder): Any two composition (principle) series of a group are isomorphic.Proof: Use Schreier since these are maximally refined.

  • HW on Section 35Dont hand in:Pages 319-321: 1, 3, 5, 7Do hand in:Pages 319-321: 2, 4, 6, 8