Problem 3: Homotopy and Fundamental Group Calculations Let X be a torus T², and consider the following loops on T²: • Loop a runs around the "longitude" of the torus. • Loop ẞ runs around the "latitude" of the torus. 1. Compute the fundamental group π1 (7², xo), where x is a base point, using generators and relations. 2. Show that any loop on 72 can be expressed as a combination of a and ẞ, and prove that the group is isomorphic to Z × Z. → 3. Consider a continuous map f: T² S¹ (where S¹ is the unit circle) that "wraps" the torus around the circle. Describe the induced map on the fundamental group f* : π₁(T²) → T1 (S1) and discuss its kernel and image. 4. Provide a graphical representation of the loops a, ẞ, and an example of a homotopy on the torus. Problem 4: Spectral Theory in Functional Analysis Let T : X → X be a bounded linear operator on a Banach space X, with the spectrum of T denoted by σ (T). 1. Prove that if T is compact, then every non-zero element of the spectrum σ (T) is an eigenvalue of T.

Problem 3: Homotopy and Fundamental Group Calculations Let X be a torus T², and consider the following loops on T²: • Loop a runs around the "longitude" of the torus. • Loop ẞ runs around the "latitude" of the torus. 1. Compute the fundamental group π1 (7², xo), where x is a base point, using generators and relations. 2. Show that any loop on 72 can be expressed as a combination of a and ẞ, and prove that the group is isomorphic to Z × Z. → 3. Consider a continuous map f: T² S¹ (where S¹ is the unit circle) that "wraps" the torus around the circle. Describe the induced map on the fundamental group f* : π₁(T²) → T1 (S1) and discuss its kernel and image. 4. Provide a graphical representation of the loops a, ẞ, and an example of a homotopy on the torus. Problem 4: Spectral Theory in Functional Analysis Let T : X → X be a bounded linear operator on a Banach space X, with the spectrum of T denoted by σ (T). 1. Prove that if T is compact, then every non-zero element of the spectrum σ (T) is an eigenvalue of T.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 25E

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9. Find all homomorphic images of the octic group.
The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
Find all subgroups of the octic group D4.
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In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.
In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.
Pls answer 1 to 3
Please do part a and b and show step by step
1.e. Consider the dihedral group (D3, ◦), ie. the symmetry group of an equilateral triangle. Determine whether (D3, ◦) has a subset of order 2.
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