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 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

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 11E: Let R be the set of all matrices of the form [abba], where a and b are real numbers. Assume that R...

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