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Let G be a finite group and A be a G-module. a) Define the group cohomology groups Hn (G, A) for n ≥ 0 and provide explicit constructions for H¹(G, A) and H² (G, A). b) Prove that H¹(G, A) classifies the equivalence classes of crossed homomorphisms from G to A. Provide a detailed proof of this classification. c) Show that H² (G, A) classifies the equivalence classes of central extensions of G by A. Explain the correspondence between cohomology classes and group extensions. d) Compute H¹(G, C×) where G is the cyclic group of order n and C× is the multiplicative group of nonzero complex numbers. Provide a complete calculation and interpretation of the result. e) Using the Lyndon-Hochschild-Serre spectral sequence, relate the cohomology of G to that of a normal subgroup N and the quotient group G/N. Provide a specific example illustrating this relationship and perform the necessary computations. Let D be a finite-dimensional division ring over its center Z(D), which is a finite field Fq. a) Prove that D is a finite field extension of Z(D) and determine its degree. Utilize Wedderburn's theorem in your analysis. b) Show that every finite division ring is commutative, thereby concluding that D is a field. Provide a detailed proof referencing relevant theorems. c) Investigate the structure of the multiplicative group D*. Determine its properties, such as being cyclic or abelian, and justify your conclusions. d) Explore the automorphism group Aut(D) of the division ring D. Describe its structure and determine whether all automorphisms are inner or if outer automorphisms exist. e) Examine the module theory over D by classifying all simple D-modules. Provide proofs for your classifications and discuss their significance in the broader context of module theory.

Let G be a finite group and A be a G-module. a) Define the group cohomology groups Hn (G, A) for n ≥ 0 and provide explicit constructions for H¹(G, A) and H² (G, A). b) Prove that H¹(G, A) classifies the equivalence classes of crossed homomorphisms from G to A. Provide a detailed proof of this classification. c) Show that H² (G, A) classifies the equivalence classes of central extensions of G by A. Explain the correspondence between cohomology classes and group extensions. d) Compute H¹(G, C×) where G is the cyclic group of order n and C× is the multiplicative group of nonzero complex numbers. Provide a complete calculation and interpretation of the result. e) Using the Lyndon-Hochschild-Serre spectral sequence, relate the cohomology of G to that of a normal subgroup N and the quotient group G/N. Provide a specific example illustrating this relationship and perform the necessary computations. Let D be a finite-dimensional division ring over its center Z(D), which is a finite field Fq. a) Prove that D is a finite field extension of Z(D) and determine its degree. Utilize Wedderburn's theorem in your analysis. b) Show that every finite division ring is commutative, thereby concluding that D is a field. Provide a detailed proof referencing relevant theorems. c) Investigate the structure of the multiplicative group D*. Determine its properties, such as being cyclic or abelian, and justify your conclusions. d) Explore the automorphism group Aut(D) of the division ring D. Describe its structure and determine whether all automorphisms are inner or if outer automorphisms exist. e) Examine the module theory over D by classifying all simple D-modules. Provide proofs for your classifications and discuss their significance in the broader context of module theory.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 11E: Find all homomorphic images of the quaternion group.
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Let G be a finite group and A be a G-module. a) Define the group cohomology groups Hn (G, A) for n ≥ 0 and provide explicit constructions for H¹(G, A) and H² (G, A). b) Prove that H¹(G, A) classifies the equivalence classes of crossed homomorphisms from G to A. Provide a detailed proof of this classification. c) Show that H² (G, A) classifies the equivalence classes of central extensions of G by A. Explain the correspondence between cohomology classes and group extensions. d) Compute H¹(G, C×) where G is the cyclic group of order n and C× is the multiplicative group of nonzero complex numbers. Provide a complete calculation and interpretation of the result. e) Using the Lyndon-Hochschild-Serre spectral sequence, relate the cohomology of G to that of a normal subgroup N and the quotient group G/N. Provide a specific example illustrating this relationship and perform the necessary computations.
Transcribed Image Text:Let G be a finite group and A be a G-module. a) Define the group cohomology groups Hn (G, A) for n ≥ 0 and provide explicit constructions for H¹(G, A) and H² (G, A). b) Prove that H¹(G, A) classifies the equivalence classes of crossed homomorphisms from G to A. Provide a detailed proof of this classification. c) Show that H² (G, A) classifies the equivalence classes of central extensions of G by A. Explain the correspondence between cohomology classes and group extensions. d) Compute H¹(G, C×) where G is the cyclic group of order n and C× is the multiplicative group of nonzero complex numbers. Provide a complete calculation and interpretation of the result. e) Using the Lyndon-Hochschild-Serre spectral sequence, relate the cohomology of G to that of a normal subgroup N and the quotient group G/N. Provide a specific example illustrating this relationship and perform the necessary computations.
Let D be a finite-dimensional division ring over its center Z(D), which is a finite field Fq. a) Prove that D is a finite field extension of Z(D) and determine its degree. Utilize Wedderburn's theorem in your analysis. b) Show that every finite division ring is commutative, thereby concluding that D is a field. Provide a detailed proof referencing relevant theorems. c) Investigate the structure of the multiplicative group D*. Determine its properties, such as being cyclic or abelian, and justify your conclusions. d) Explore the automorphism group Aut(D) of the division ring D. Describe its structure and determine whether all automorphisms are inner or if outer automorphisms exist. e) Examine the module theory over D by classifying all simple D-modules. Provide proofs for your classifications and discuss their significance in the broader context of module theory.
Transcribed Image Text:Let D be a finite-dimensional division ring over its center Z(D), which is a finite field Fq. a) Prove that D is a finite field extension of Z(D) and determine its degree. Utilize Wedderburn's theorem in your analysis. b) Show that every finite division ring is commutative, thereby concluding that D is a field. Provide a detailed proof referencing relevant theorems. c) Investigate the structure of the multiplicative group D*. Determine its properties, such as being cyclic or abelian, and justify your conclusions. d) Explore the automorphism group Aut(D) of the division ring D. Describe its structure and determine whether all automorphisms are inner or if outer automorphisms exist. e) Examine the module theory over D by classifying all simple D-modules. Provide proofs for your classifications and discuss their significance in the broader context of module theory.
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