2. Let G be a group. Recall from Homework 4 Supplementary Problem 1 that thecenter of G is defined to beZ(G) = { a = G | ag = ga for all g = G}.(a) Show that if X C G is a generating set for G, thenZ(G) = { a G | ax = xa for all x € X}.(b) Find the centers of Sn, Dn, and GLn. (Hint for GLn: There is a theorem fromlinear algebra that every invertible matrix can be written as a product of ele-mentary matrices. In other words, GLn is generated by elementary matrices.)

Answered: Image /qna-images/answer/33e84af2-a225-428c-99e5-4cbb07d76eb6.jpg

Answered: center of G is defined to be Z(G) = {a…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

See similar textbooks

Similar questions

Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.
Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. b. in Exercise 36 of section 3.1. c. in Exercise 35 of section 3.1. d., the general linear group of order over. Exercise 34 of section 3.1. Let be the set of eight elements with identity element and noncommutative multiplication given by for all in (The circular order of multiplication is indicated by the diagram in Figure .) Given that is a group of order , write out the multiplication table for . This group is known as the quaternion group. Exercise 36 of section 3.1 Consider the matrices in , and let . Given that is a group of order 8 with respect to multiplication, write out a multiplication table for. Exercise 35 of section 3.1. A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging the rows one or more times (that is, by permuting the rows). For the permutation matrices are and the five matrices. Given that is a group of order with respect to matrix multiplication, write out a multiplication table for .
Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
Find two groups of order 6 that are not isomorphic.
9. Suppose that and are subgroups of the abelian group such that . Prove that .
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40
2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.
9. Find all homomorphic images of the octic group.

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