Please answer the question shown in the image below: (Hint: the group is isomorphic to the group C) **Problem (18)**Let $ H $ be the set of all matrices in $ GL_2(\mathbb{R}) $ of the form \[\begin{pmatrix} a & -b \\ b & a \end{pmatrix}.\]Show that $ H $ is a subgroup of $ GL_2(\mathbb{R}) $. The group $ H $ is isomorphic to what well-known group? Prove your answer correct. **Explanation:**This problem involves linear algebra and group theory. It asks you to demonstrate that a particular set of matrices forms a subgroup of the general linear group $ GL_2(\mathbb{R}) $, which is the group of invertible 2x2 matrices with real number entries. Then, it asks you to identify a well-known group that is isomorphic to this set and prove the isomorphism.1. **Definition of Subgroup**: To show that $ H $ is a subgroup, you need to verify: - The identity matrix is in $ H $. - $ H $ is closed under matrix multiplication. - Every matrix in $ H $ has an inverse that is also in $ H $.2. **Matrix Form and Isomorphism**: - The matrices in $ H $ are of a specific form. They resemble rotations in the plane, which are linked to complex numbers. - The group $ H $ is likely isomorphic to the circle group $ S^1 $ or $ U(1) $, which represents the set of complex numbers with absolute value 1, under multiplication.3. **Proving Isomorphism**: - Show a bijective homomorphism between $ H $ and the known group. - Verify the homomorphism properties and inverse mapping.This problem combines abstract algebra with matrix transformations, highlighting connections with other areas like complex numbers and geometry.

Answered: Image /qna-images/answer/9171e6c8-d825-4b01-aab8-988a1c84eb8e.jpg

Answered: а H be the set of all matrices in…

Math

Advanced Math

а H be the set of all matrices in GL2(R) of the form b. a

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

Q: Lagrange's theorem, in group theory, a part of mathematics, states that for any finite group G, the…

Q: Question 6 If G, H and K are groups, and fo: G→H and f₁: HK are isomorphisms, show that fi fo: GK is…

Q: How would you use the first isomorphic theorem to prove that G/K is isomorphic to G?

A: The objective of the question is to use the first isomorphism theorem to prove that the quotient…

Q: Let m and n be relatively prime positive integers. Prove that the order of the element (1, 1) of the…

A: To prove ℤm×ℤn≅ℤmn ⇔ gcd(m,n)=1 Part 1) Proof of 'only if' part ℤm×ℤn≅ℤmn ⇒ℤm×ℤn is cyclic. Let…

Q: Theorem 2.3.1: Let (G, o) be a group, ae G, and m, ne N, then the powers of a satisf the following…

A: If G is a set and * be the operation on the elements of the set of G such that (G , *) is a group…

Q: Explicitly construct the Galois group for r- 4r² + 2 over Q. To which group is this isomorphic?

Q: Let ø be a homomorphism from a group G to a group G'. Prove that ker øis a subgroup of G.

Q: For any group element a and any integer k, show that C(a) C C(a*). Use this fact to complete the…

A: To prove: Ca⊆Cak for any group element a and any integer k. Also, To Complete the statement "In a…

Q: Dicn = (r,s | 2,¹s-2, rsrs-1). (a) Prove that Dic, has at most 4n elements. (In fact, it has exactly…

Q: Compute the left regular representation of the group G = {e,a, b} given by the group table below by…

Q: Let G be a group isomorphic to Zn, O Zn, O ...OZ. Let x be the product of all elements in G.…

A: Given: G is a group isomorphic to ℤn1⊕ℤn1⊕....⊕ℤnk and x is the product of all elements in G.To…

Q: Theorem 3.34: Let (G, *) be a group, then (1) {e}AG. (2) GAG (3) Cent(G) AG. Proof: (H.W.)

Q: Give an example to show that there exists a group with elements a and b such that la] = 0, \b| = ∞,…

Q: Prove or disproe: If C is a groun and a hsC then (ab)-1 h-1a-1

Q: PROPOSITION 9.2. Let f :G G' be an isomorphism of groups. For any element a of G, we have the…

Q: If F is a reflection in the dihedral group D, find all elements X in D, such that X? = F and all…

Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

Q: Let G be any group and p: G -G, (g) =g1 Vg eG. Then: (a) o is a homomorphism (b) o is a homomorphism…

Q: . Let G be a group and a e G. An element be G is called a conjugate of a if there exists an element…

Q: 1.e. Consider the dihedral group (D3, ◦), ie. the symmetry group of an equilateral triangle.…

A: Symmetry group of an equilateral triangle have subgroup of order 2 and it is isomorphic to Z2

Q: Consider the group (U(20), '20). 1. What is the order of the group? 2. What is the inverse of 7,9…

Q: Question 4 Show that R* x R* is not isomorphic to C*. (R* is R\ {0}, and C* is C\{0}. In both cases,…

A: Show that is not isomorphic to . ( is , and is . In both cases, the group operation is…

Q: Prove that the additive group L is isomorphic to the multiplicative group of nonzero elements in

Q: Given that ø is a isomorphism from a group G under addition to a group G under addition, convert…

Q: Find the orders of each of the elements of U(14). Show that it is cyclic and find all its…

A: Given :- The orders of each of the elements of U14. To Determine :- It is cyclic and all its…

Q: Lagrange's theorem, in group theory, a part of mathematics, states that for any finite group G, the…

Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.

Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

Q: State whether each of the following statements is true or false. Then give an explanation, proof o…

A: considering a=a-1 and b=b-1 as a=a-1 and b=b-1 the inverse of ba is given by ba-1=a-1b-1…

Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

Q: Compute the left regular representation of the group G = {e, a, b} given by the group table below by…

A: Inverse element : Suppose G is a group and a∈G then their exit an inverse element b(let) such that…

Q: Question 2 Let G be a group. (i) For any a E G, suppose b and c are both inverses of a. By…

Q: Write out the Cayley table for the dihedral group D6 = {1,r,r2,s,sr,sr2}. (In general, we apply the…

Q: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…

A: We have to solve given problem:

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a+bv2 : a,…

Q: Let C be a group and let a be an element of C Prove that = . Please be clear with theorems ana…

Q: Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload…

Q: Let H = (a) be a cyclic group. Define f: Z → H by f(k) = a*. Prove that f is a surjective…

Question

Please answer the question shown in the image below: (Hint: the group is isomorphic to the group C)

**Problem (18)**

Let $ H $ be the set of all matrices in $ GL_2(\mathbb{R}) $ of the form

\[
\begin{pmatrix}
a & -b \\
b & a
\end{pmatrix}.
\]

Show that $ H $ is a subgroup of $ GL_2(\mathbb{R}) $. The group $ H $ is isomorphic to what well-known group? Prove your answer correct.

**Explanation:**

This problem involves linear algebra and group theory. It asks you to demonstrate that a particular set of matrices forms a subgroup of the general linear group $ GL_2(\mathbb{R}) $, which is the group of invertible 2x2 matrices with real number entries. Then, it asks you to identify a well-known group that is isomorphic to this set and prove the isomorphism.

1. **Definition of Subgroup**: To show that $ H $ is a subgroup, you need to verify:
- The identity matrix is in $ H $.
- $ H $ is closed under matrix multiplication.
- Every matrix in $ H $ has an inverse that is also in $ H $.

2. **Matrix Form and Isomorphism**:
- The matrices in $ H $ are of a specific form. They resemble rotations in the plane, which are linked to complex numbers.
- The group $ H $ is likely isomorphic to the circle group $ S^1 $ or $ U(1) $, which represents the set of complex numbers with absolute value 1, under multiplication.

3. **Proving Isomorphism**:
- Show a bijective homomorphism between $ H $ and the known group.
- Verify the homomorphism properties and inverse mapping.

This problem combines abstract algebra with matrix transformations, highlighting connections with other areas like complex numbers and geometry.

Expert Solution

Step 1

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Show that (ℤ,∗)?ℎ??? r ∗s = (r +s)−(r ∙ s)??? ??? r,s ∈ ℤ is group using variable r, s and t.
Explain in detail if the following statement is true or false: " Suppose a group G acts on a set X; then the collection of stabilizers {Gx : x ∈ X} forms apartition of G"
Which answer is correct?
Consider the group G- (x E R such that x 0} under the binary operation The inverse element x is: -4/x 1/4x 2/x 4/x
Help with this please
I have the solution to this question. I just need 2-3 examples of this question.
Express the lengths a and b shown in the figure in terms of 0. Do not solve for a and b. 0 24 b a
G={2.3" | m,ne Z} TASK 2: Prove that if G is a group and a, b = G, then (1) o(a¹ba) = o(b) (ii) o(ab) = o(ba)
q17: The inverse of the element 5 in the group{1,3,5,7} is A)7 B)1 C)5 D)3
a.) Explain in your own words what this problem is asking. b.) Explain the meaning of any notation used in the problem and in your solution. c.) Describe the mathematical concept(s) that appear to be foundational to this problem. d.) Justified solution to or proof of the problem.
5. To what group is GL2(Z2) isomorphic to?