3. Consider a regular octagon (eight-sided regular polygon, like a stop sign). Number its vertices 1 through 8, going clockwise. Let G denote the group of symmetries of the octagon-it has order 16, with 8 rotations and 8 reflections. (a) Describe each of the 16 elements of G (identify each rotation by its angle, and each reflection by giving its line of symmetry). Then give the corresponding element of Sg (according to how the vertices are permuted). (b) Let H consist of the cyclic subgroup generated by rotation by 90°. Determine whether H is normal in G, and give a convincing explanation for your conclusion. (c) Let K consist of the subgroup containing the identity, rotation by 180°, reflection across the line between 1 and 5, and reflection across the line between 3 and 7. (You do not need to verify that it is a subgroup-trust me on this.) Determine whether K is normal in G, and give a convincing explanation for your conclusion. (d) More generally, suppose you have a d-sided regular polygon, whose symmetry group G has 2d elements (d rotations and d reflections). Let H be any subgroup of G. Prove that either H consists entirely of rotations, or half the elements of H are rotations and half are reflections.
3. Consider a regular octagon (eight-sided regular polygon, like a stop sign). Number its vertices 1 through 8, going clockwise. Let G denote the group of symmetries of the octagon-it has order 16, with 8 rotations and 8 reflections. (a) Describe each of the 16 elements of G (identify each rotation by its angle, and each reflection by giving its line of symmetry). Then give the corresponding element of Sg (according to how the vertices are permuted). (b) Let H consist of the cyclic subgroup generated by rotation by 90°. Determine whether H is normal in G, and give a convincing explanation for your conclusion. (c) Let K consist of the subgroup containing the identity, rotation by 180°, reflection across the line between 1 and 5, and reflection across the line between 3 and 7. (You do not need to verify that it is a subgroup-trust me on this.) Determine whether K is normal in G, and give a convincing explanation for your conclusion. (d) More generally, suppose you have a d-sided regular polygon, whose symmetry group G has 2d elements (d rotations and d reflections). Let H be any subgroup of G. Prove that either H consists entirely of rotations, or half the elements of H are rotations and half are reflections.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I need help with question 3
Transcribed Image Text:3. Consider a regular octagon (eight-sided regular polygon, like a stop sign). Number its vertices
1 through 8, going clockwise. Let G denote the group of symmetries of the octagon it has order 16,
with 8 rotations and 8 reflections.
(a) Describe each of the 16 elements of G (identify each rotation by its angle, and each reflection
by giving its line of symmetry). Then give the corresponding element of Sg (according to how the
vertices are permuted).
(b)
Let H consist of the cyclic subgroup generated by rotation by 90°. Determine whether H
is normal in G, and give a convincing explanation for your conclusion.
(c)
Let K consist of the subgroup containing the identity, rotation by 180°, reflection across
the line between 1 and 5, and reflection across the line between 3 and 7. (You do not need to verify
that it is a subgroup trust me on this.) Determine whether K is normal in G, and give a convincing
explanation for your conclusion.
(d)
More generally, suppose you have a d-sided regular polygon, whose symmetry group G has
2d elements (d rotations and d reflections). Let H be any subgroup of G. Prove that either H consists
entirely of rotations, or half the elements of H are rotations and half are reflections.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 128 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
