No without AI fast before 4:45 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 andcharacter 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 matricesrepresenting the group elements under each representation.d) Using the representations from part (c), decompose the regular representation of G into itsirreducible components. Explain each step of your decomposition.

Answered: Image /qna-images/answer/06127d1b-6606-4366-b683-46329b77f41e.jpg

Answered: Let G be a finite group of order 21,…

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.

See similar textbooks

Similar questions

Prove that the set of all complex numbers that have absolute value forms a group with respect to multiplication.
Need detailed solutions to each parts as ap
All parts fast
under multiplication and let G be the group of all real 2x 2 matrices of -- Let G be the group of all non-zero complex numbers under multiplication and let G be the eroup of all real 2× 2 matrices of a b -b a the form where a and b are not both zero under matrix nlir plication. Show that G and G are isomorphic by exhibiting an isomorphism of G onto G.
Let C* be the group of nonzero complex numbers under multiplica- tion and let H = {a+bi E C* | a² + b² = 1}. Give a geometric de- scription of the coset (3 + 4i)H. Give a geometric description of the coset (c+ di)H.
Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the system (Zeven, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zeven, +) a semi-group? Explain Is (Zeven, +) a group? Explain
Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system (Zodd, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zodd, +) a semi-group? Explain Is (Zodd, +) a group? Explain
2. Let H be the set of 2 × 2 matrices of the form H a b = { (- / + d ) | a,b ≤ c } a where a denotes the complex conjugate of a complex number a. Show that H, together with matrix addition and multiplication, forms a ring. (You may assume that M₂(C) is a ring.)
Prove that the set of all 2 x 2 matrices with entries from R and determinant+1 is a group under matrix multiplication.
Show that the mapping Φ(a + bi) = a - bi is an automorphism of the group of complex numbers under addition. Show that Φ preserves complex multiplication as well—that is, Φ(xy) = Φ(x)Φ(y)
(1) Find all left, right and two-sided ideals of the ring of n x n complex matrices.
Let R be a ring and let M2(R) be the ring of 2 x 2 matrices with entriesfrom R. Explain why these two rings have the same characteristic.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS