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Let H be the group given in Exercise 17 of Section 3.3 , and let S ( A ) be as given in Example 2 of this section. Find an isomorphism from H to S ( A ) . Sec. 3.3 , # 17 ≫ Consider the set of matrices H = { I 2 , M 1 , M 2 , M 3 , M 4 , M 5 } , where I 2 = [ 1 0 0 1 ] , M 1 = [ 1 0 − 1 − 1 ] , M 2 = [ 0 1 − 1 − 1 ] M 3 = [ − 1 − 1 1 0 ] , M 4 = [ − 1 − 1 0 1 ] , M 5 = [ 0 1 1 0 ] Show that H is a subgroup of G L ​ ( 2 , ℝ ) , the general linear group of order 2 over ℝ .

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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 3.5, Problem 5E
Textbook Problem
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Let H be the group given in Exercise 17 of Section 3.3 , and let S ( A ) be as given in Example 2 of this section. Find an isomorphism from H to S ( A ) .

Sec. 3.3 , # 17

Consider the set of matrices H = { I 2 , M 1 , M 2 , M 3 , M 4 , M 5 } , where

I 2 = [ 1 0 0 1 ] , M 1 = [ 1 0 1 1 ] , M 2 = [ 0 1 1 1 ]

M 3 = [ 1 1 1 0 ] , M 4 = [ 1 1 0 1 ] , M 5 = [ 0 1 1 0 ]

Show that H is a subgroup of G L ( 2 , ) , the general linear group of order 2 over .

To determine

To find: An isomorphism from H to S(A).

Explanation of Solution

Given information:

H={I2,M1,M2,M3,M4,M5} where I2=[1001],M1=[1011],M2=[0111],M3=[1110],M4=[1101],M5=[0110] under multiplication.

S(A)={IA,ρ,τ,σ,γ,δ} where

IA={IA(1)=1IA(2)=2IA(3)=3, σ={σ(1)=2σ(2)=1σ(3)=3, ρ={ρ(1)=2ρ(2)=3ρ(3)=1

γ={γ(1)=3γ(2)=2γ(3)=1, τ={τ(1)=3τ(2)=1τ(3)=2, δ={δ(1)=1δ(2)=3δ(3)=2

under multiplication.

Formula used:

A mapping ϕ:GG' is isomorphism if

1) ϕ is one-to-one correspondence from G to G'.

2) ϕ(xy)=ϕ(x)ϕ(y) for all x,y in G.

Calculation:

Let the first group H={I2,M1,M2,M3,M4,M5} under multiplication and second group S(A)={IA,ρ,τ,σ,γ,δ} under multiplication.

A mapping ϕ:HS(A) defined by ϕ(I2)=IA,ϕ(M1)=σ,ϕ(M2)=ρ,ϕ(M3)=τ,ϕ(M4)=γ,ϕ(M5)=δ.

For every element of H, there is a different image in S(A), so ϕ:HS(A) is one-one. That is,

abϕ(a)ϕ(b), so ϕ:HS(A) is one-one.

For every element of S(A), there is preimage in H, so ϕ:HS(A) is onto. That is,

xS(A),aH such that ϕ(a)=x

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Elements Of Modern Algebra
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Ch. 3.1 - True or False Label each of the following...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises , decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - In Exercises and, the given table defines an...Ch. 3.1 - In Exercises 15 and 16, the given table defines an...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises 1724, let the binary operation be...Ch. 3.1 - In Exercises 1724, let the binary operation be...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises 2532, decide whether each of the...Ch. 3.1 - In Exercises 2532, decide whether each of the...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - a. Let G={ [ a ][ a ][ 0 ] }n. Show that G is a...Ch. 3.1 - 34. Let be the set of eight elements with...Ch. 3.1 - 35. A permutation matrix is a matrix that can be...Ch. 3.1 - Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[...Ch. 3.1 - Prove or disprove that the set of all diagonal...Ch. 3.1 - 38. Let be the set of all matrices in that have...Ch. 3.1 - 39. Let be the set of all matrices in that have...Ch. 3.1 - 40. Prove or disprove that the set in Exercise ...Ch. 3.1 - 41. Prove or disprove that the set in Exercise ...Ch. 3.1 - 42. For an arbitrary set , the power set was...Ch. 3.1 - Write out the elements of P(A) for the set A={...Ch. 3.1 - Let A={ a,b,c }. Prove or disprove that P(A) is a...Ch. 3.1 - 45. Let . Prove or disprove that is a group with...Ch. 3.1 - In Example 3, the group S(A) is nonabelian where...Ch. 3.1 - 47. Find the additive inverse of in the given...Ch. 3.1 - Find the additive inverse of [ [ 2 ][ 3 ][ 4 ][ 1...Ch. 3.1 - 49. 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If is a cyclic group of order and ...Ch. 3.4 - Suppose that a and b are elements of finite order...Ch. 3.4 - Suppose that a is an element of order m in a group...Ch. 3.4 - Exercises 38. Assume that is a cyclic group of...Ch. 3.4 - Suppose a is an element of order mn in a group G,...Ch. 3.4 - Exercises 40. Prove or disprove: If every...Ch. 3.4 - Let G be an abelian group. Prove that the set of...Ch. 3.4 - Let d be a positive integer and (d) the Euler...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Prove that if is an isomorphism from the group G...Ch. 3.5 - Let G1, G2, and G3 be groups. Prove that if 1 is...Ch. 3.5 - Exercises 3. Find an isomorphism from the additive...Ch. 3.5 - Let G=1,i,1,i under multiplication, and let G=4=[...Ch. 3.5 - Let H be the group given in Exercise 17 of Section...Ch. 3.5 - Exercises 6. Find an isomorphism from the additive...Ch. 3.5 - Find an isomorphism from the additive group to...Ch. 3.5 - Exercises 8. Find an isomorphism from the group ...Ch. 3.5 - Exercises 9. Find an isomorphism from the...Ch. 3.5 - Exercises 10. Find an isomorphism from the...Ch. 3.5 - The following set of matrices [ 1001 ], [ 1001 ],...Ch. 3.5 - Exercises 12. Prove that the additive group of...Ch. 3.5 - Consider the groups given in Exercise 12. Find an...Ch. 3.5 - Consider the additive group of real numbers....Ch. 3.5 - Consider the additive group of real numbers....Ch. 3.5 - Exercises 16. Assume that the nonzero complex...Ch. 3.5 - Let G be a group. Prove that G is abelian if and...Ch. 3.5 - Exercises 18. Suppose and let be defined by ....Ch. 3.5 - According to Exercise of Section, If n is a prime,...Ch. 3.5 - For each a in the group G, define a mapping ta:GG...Ch. 3.5 - For a fixed group G, prove that the set of all...Ch. 3.5 - Exercises 22. Let be a finite cyclic group of...Ch. 3.5 - Exercises 23. Assume is a (not necessarily...Ch. 3.5 - Let G be as in Exercise 23. Suppose also that ar...Ch. 3.5 - Exercises 25. Let be the multiplicative group of...Ch. 3.5 - Exercises 26. Use the results of Exercises and ...Ch. 3.5 - Exercises 27. Consider the additive groups , , and...Ch. 3.5 - Exercises 28. Let , , , and be groups with...Ch. 3.5 - Prove that any cyclic group of finite order n is...Ch. 3.5 - Exercises 30. For an arbitrary positive integer,...Ch. 3.5 - Prove that any infinite cyclic group is isomorphic...Ch. 3.5 - Let H be the group 6 under addition. Find all...Ch. 3.5 - Suppose that G and H are isomorphic groups. Prove...Ch. 3.5 - Exercises 34. Prove that if and are two groups...Ch. 3.5 - Exercises 35. Prove that any two groups of order ...Ch. 3.5 - Exercises 36. Exhibit two groups of the same...Ch. 3.5 - Let be an isomorphism from group G to group H....Ch. 3.5 - Exercises 38. If and are groups and is an...Ch. 3.5 - Suppose that is an isomorphism from the group G...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Each of the following rules determines a mapping...Ch. 3.6 - Each of the following rules determines a mapping ...Ch. 3.6 - 3. Consider the additive groups of real numbers...Ch. 3.6 - Consider the additive group and the...Ch. 3.6 - 5. Consider the additive group and define...Ch. 3.6 - Consider the additive groups 12 and 6 and define...Ch. 3.6 - Consider the additive groups 8 and 4 and define...Ch. 3.6 - 8. Consider the additive groups and . Define by...Ch. 3.6 - 9. Let be the additive group of matrices over...Ch. 3.6 - Rework exercise 9 with G=GL(2,), the general...Ch. 3.6 - 11. Let be , and let be the group of nonzero real...Ch. 3.6 - Consider the additive group of real numbers. Let ...Ch. 3.6 - Find an example of G, G and such that G is a...Ch. 3.6 - 14. Let be a homomorphism from the group to the...Ch. 3.6 - 15. Prove that on a given collection of groups,...Ch. 3.6 - 16. Suppose that and are groups. If is a...Ch. 3.6 - 17. Find two groups and such that is a...Ch. 3.6 - Suppose that is an epimorphism from the group G...Ch. 3.6 - 19. Let be a homomorphism from a group to a group...Ch. 3.6 - 20. If is an abelian group and the group is a...Ch. 3.6 - 21. Let be a fixed element of the multiplicative...Ch. 3.6 - 22. With as in Exercise , show that , and describe...Ch. 3.6 - Assume that is a homomorphism from the group G to...Ch. 3.6 - 24. Assume that the group is a homomorphic image...Ch. 3.6 - Let be a homomorphism from the group G to the...

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