1. The regular representation of a finite group G is a pair (Vreg, Dreg). Vreg is a vector space and Dreg is a homomorphism. (a) What is the dimension of Vreg? (b) Describe a basis for Vreg and give a formula for Dreg. Hence explain why the homo- morphism property is satisfied by Dreg. (c) Prove that the character ✗reg (g) defined by tr Dreg (g) is zero if g is not the identity element of the group. (d) A finite group of order 60 has five irreducible representations R1, R2, R3, R4, R5. R₁ is the trivial representation. R2, R3, R4 have dimensions (3,3,4) respectively. What is the dimension of R5? Explain how your solution is related to the decomposition of the regular representation as a direct sum of irreducible representations (You can assume without proof the properties of this decomposition which have been explained in class and in the lecture notes). (e) A group element has characters in the irreducible representations R2, R3, R4 given as R3 R2 (g) = -1 X³ (g) = −1 ; XR4 (g) = 0 Calculate XR5 (g), explaining your reasoning.

1. The regular representation of a finite group G is a pair (Vreg, Dreg). Vreg is a vector space and Dreg is a homomorphism. (a) What is the dimension of Vreg? (b) Describe a basis for Vreg and give a formula for Dreg. Hence explain why the homo- morphism property is satisfied by Dreg. (c) Prove that the character ✗reg (g) defined by tr Dreg (g) is zero if g is not the identity element of the group. (d) A finite group of order 60 has five irreducible representations R1, R2, R3, R4, R5. R₁ is the trivial representation. R2, R3, R4 have dimensions (3,3,4) respectively. What is the dimension of R5? Explain how your solution is related to the decomposition of the regular representation as a direct sum of irreducible representations (You can assume without proof the properties of this decomposition which have been explained in class and in the lecture notes). (e) A group element has characters in the irreducible representations R2, R3, R4 given as R3 R2 (g) = -1 X³ (g) = −1 ; XR4 (g) = 0 Calculate XR5 (g), explaining your reasoning.

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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