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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 G be a finite group of order 21, and let C denote the field of complex numbers. a) Determine all possible irreducible complex representations of G. Provide their dimensions and character tables. b) Prove that every irreducible representation of G is one-dimensional or three-dimensional. Justify your reasoning based on the structure of G. c) Construct explicitly the irreducible representations identified in part (a). Provide matrices representing the group elements under each representation. d) Using the representations from part (c), decompose the regular representation of G into its irreducible components. Explain each step of your decomposition.

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 G be a finite group of order 21, and let C denote the field of complex numbers. a) Determine all possible irreducible complex representations of G. Provide their dimensions and character tables. b) Prove that every irreducible representation of G is one-dimensional or three-dimensional. Justify your reasoning based on the structure of G. c) Construct explicitly the irreducible representations identified in part (a). Provide matrices representing the group elements under each representation. d) Using the representations from part (c), decompose the regular representation of G into its irreducible components. Explain each step of your decomposition.

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 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.
Let G be a finite group of order 21, and let C denote the field of complex numbers. a) Determine all possible irreducible complex representations of G. Provide their dimensions and character tables. b) Prove that every irreducible representation of G is one-dimensional or three-dimensional. Justify your reasoning based on the structure of G. c) Construct explicitly the irreducible representations identified in part (a). Provide matrices representing the group elements under each representation. d) Using the representations from part (c), decompose the regular representation of G into its irreducible components. Explain each step of your decomposition.
Transcribed Image Text:Let G be a finite group of order 21, and let C denote the field of complex numbers. a) Determine all possible irreducible complex representations of G. Provide their dimensions and character tables. b) Prove that every irreducible representation of G is one-dimensional or three-dimensional. Justify your reasoning based on the structure of G. c) Construct explicitly the irreducible representations identified in part (a). Provide matrices representing the group elements under each representation. d) Using the representations from part (c), decompose the regular representation of G into its irreducible components. Explain each step of your decomposition.
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