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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 non-abelian simple group of order 60. a) Determine all possible isomorphism types of G. Justify your conclusion using the classification of simple groups of small order. b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure, including any inner and outer automorphisms. c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms. Use this to discuss the simplicity of Aut (G). d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this action is transitive and justify your answer.

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 non-abelian simple group of order 60. a) Determine all possible isomorphism types of G. Justify your conclusion using the classification of simple groups of small order. b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure, including any inner and outer automorphisms. c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms. Use this to discuss the simplicity of Aut (G). d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this action is transitive and justify your answer.

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 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.
Let G be a finite non-abelian simple group of order 60. a) Determine all possible isomorphism types of G. Justify your conclusion using the classification of simple groups of small order. b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure, including any inner and outer automorphisms. c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms. Use this to discuss the simplicity of Aut (G). d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this action is transitive and justify your answer.
Transcribed Image Text:Let G be a finite non-abelian simple group of order 60. a) Determine all possible isomorphism types of G. Justify your conclusion using the classification of simple groups of small order. b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure, including any inner and outer automorphisms. c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms. Use this to discuss the simplicity of Aut (G). d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this action is transitive and justify your answer.
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