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Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination. Let G be a finite group of order 56. a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification using Sylow's theorems. b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal in G, and subsequently show that G contains a normal subgroup of order 8. c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss whether G is necessarily abelian or non-abelian based on your classification.

Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination. Let G be a finite group of order 56. a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification using Sylow's theorems. b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal in G, and subsequently show that G contains a normal subgroup of order 8. c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss whether G is necessarily abelian or non-abelian based on your classification.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 17E
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Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination.
Transcribed Image Text:Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination.
Let G be a finite group of order 56. a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification using Sylow's theorems. b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal in G, and subsequently show that G contains a normal subgroup of order 8. c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss whether G is necessarily abelian or non-abelian based on your classification.
Transcribed Image Text:Let G be a finite group of order 56. a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification using Sylow's theorems. b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal in G, and subsequently show that G contains a normal subgroup of order 8. c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss whether G is necessarily abelian or non-abelian based on your classification.
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