Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
6. The dihedral group D5 of isometries of a regular pentagon has elements 4 {e‚r, r², r³, r², x, rx, r²x,r³x, r²x} where r is a rotation by angle 2π/5 and x, rx, r²x, r³x, r¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, x has order 2 and xr = (i) Show by induction on n that xr² = r¯nx for all n ≥ 0. (Note that r -1 = r².) r¼x. 1 (ii) Show that the subset {e, r, r², r³, r4} is a normal subgroup.

6. The dihedral group D5 of isometries of a regular pentagon has elements 4 {e‚r, r², r³, r², x, rx, r²x,r³x, r²x} where r is a rotation by angle 2π/5 and x, rx, r²x, r³x, r¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, x has order 2 and xr = (i) Show by induction on n that xr² = r¯nx for all n ≥ 0. (Note that r -1 = r².) r¼x. 1 (ii) Show that the subset {e, r, r², r³, r4} is a normal subgroup.

Elementary Geometry for College Students
Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
Chapter2: Parallel Lines
Section2.6: Symmetry And Transformations
Problem 28E
See similar textbooks
icon
Related questions
Question
6. The dihedral group D5 of isometries of a regular pentagon has elements 4 {e‚r, r², r³, r², x, rx, r²x,r³x, r²x} where r is a rotation by angle 2π/5 and x, rx, r²x, r³x, r¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, x has order 2 and xr = (i) Show by induction on n that xr² = r¯nx for all n ≥ 0. (Note that r -1 = r².) r¼x. 1 (ii) Show that the subset {e, r, r², r³, r4} is a normal subgroup.
Transcribed Image Text:6. The dihedral group D5 of isometries of a regular pentagon has elements 4 {e‚r, r², r³, r², x, rx, r²x,r³x, r²x} where r is a rotation by angle 2π/5 and x, rx, r²x, r³x, r¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, x has order 2 and xr = (i) Show by induction on n that xr² = r¯nx for all n ≥ 0. (Note that r -1 = r².) r¼x. 1 (ii) Show that the subset {e, r, r², r³, r4} is a normal subgroup.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Elementary Geometry for College Students
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Holt Mcdougal Larson Pre-algebra: Student Edition…
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS