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. Consider the category of R-modules, where R is a commutative ring with unity, and let 0 → ABC → 0 be a short exact sequence of R-modules. a) Prove that the sequence is exact if and only if Im(ƒ) = ker(g), ƒ is injective, and g is surjective. Provide a detailed explanation of each implication. b) Assume R is a Noetherian ring and A and C are finitely generated R-modules. Prove that B is also finitely generated. Use the properties of Noetherian rings in your proof. c) Define the notion of a projective module and prove that if A is a projective R-module, then the short exact sequence splits. Provide a comprehensive proof of the splitting. d) Explore the Ext functor by computing Ext (C, A) for the given exact sequence. Explain the relationship between the exact sequence and the Ext group, and determine the conditions under which the sequence splits.

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 22E: 22. Let be a ring with finite number of elements. Show that the characteristic of divides .