3. (Third Isomorphism Theorem) We wish to prove the Third Isomorphism Theorem, which states hat if HG, NG, and N H, then G/H (G/N)/(H/N). In the below diagram, let π, π', and πΗ denote the associated quotient homomorphisms. (G/N)/(H/N) k GG/N ΠΗ G/H a) Use the Correspondence Theorem to show that for NG and HN, HGH/NG/N. b) Show that ker (π' οπ) = Η. c) Use one of our other isomorphism theorems to show that there exists an isomor- phism in the above diagram that makes the diagram commute. 4. (Quotient group computations) Let GZ/60Z, H=6Z/60Z = ([6]) ≤ Z/60Z, and N=4Z/60Z ([4]) ≤ Z/60Z. a) State the Second Isomorphism Theorem in terms of additive notation. b) Compute HΩΝ. c) Compute H + N. d) Show that (H + N)/N = H/(HON) without using the Second Isomorphism Theorem.
3. (Third Isomorphism Theorem) We wish to prove the Third Isomorphism Theorem, which states hat if HG, NG, and N H, then G/H (G/N)/(H/N). In the below diagram, let π, π', and πΗ denote the associated quotient homomorphisms. (G/N)/(H/N) k GG/N ΠΗ G/H a) Use the Correspondence Theorem to show that for NG and HN, HGH/NG/N. b) Show that ker (π' οπ) = Η. c) Use one of our other isomorphism theorems to show that there exists an isomor- phism in the above diagram that makes the diagram commute. 4. (Quotient group computations) Let GZ/60Z, H=6Z/60Z = ([6]) ≤ Z/60Z, and N=4Z/60Z ([4]) ≤ Z/60Z. a) State the Second Isomorphism Theorem in terms of additive notation. b) Compute HΩΝ. c) Compute H + N. d) Show that (H + N)/N = H/(HON) without using the Second Isomorphism Theorem.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 4E: Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an...
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3. (Third Isomorphism Theorem) We wish to prove the Third Isomorphism Theorem, which states hat if HG, NG, and N H, then G/H (G/N)/(H/N). In the below diagram, let π, π', and πΗ denote the associated quotient homomorphisms.
(G/N)/(H/N)
k
GG/N
ΠΗ
G/H
a) Use the Correspondence Theorem to show that for NG and HN, HGH/NG/N.
b) Show that ker (π' οπ) = Η.
c) Use one of our other isomorphism theorems to show that there exists an isomor- phism in the above diagram that makes the diagram commute.
4. (Quotient group computations) Let GZ/60Z, H=6Z/60Z = ([6]) ≤ Z/60Z, and N=4Z/60Z ([4]) ≤ Z/60Z.
a) State the Second Isomorphism Theorem in terms of additive notation.
b) Compute HΩΝ. c) Compute H + N.
d) Show that (H + N)/N = H/(HON) without using the Second Isomorphism Theorem.
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