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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
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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.
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