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6. The dihedral group D5 of isometries of a regular pentagon has elements {e,r,r²,r³, r²¹, x, rx, r²x, r³x, r²¹x} where r is a rotation by angle 27/5 and x, rx, r²x, r³x, ¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, 2 has order 2 and ær=r¹²x. (i) Show by induction on n that r = ra for all n ≥ 0. (Note that r¹=r¹.) 3 (ii) Show that the subset {e,r,r², r³,r¹} is a normal subgroup.

6. The dihedral group D5 of isometries of a regular pentagon has elements {e,r,r²,r³, r²¹, x, rx, r²x, r³x, r²¹x} where r is a rotation by angle 27/5 and x, rx, r²x, r³x, ¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, 2 has order 2 and ær=r¹²x. (i) Show by induction on n that r = ra for all n ≥ 0. (Note that r¹=r¹.) 3 (ii) Show that the subset {e,r,r², r³,r¹} is a normal subgroup.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6. The dihedral group D5 of isometries of a regular pentagon has elements {e,r,r², r³, r¹, x, rx, r²x, r³x, r²¹x} where r is a rotation by angle 27/5 and x, rx, r²x, r³x, ¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, 2 has order 2 and ær = r¹²x. (i) Show by induction on n that r = rr for all n ≥0. (Note that r¹=r¹.) 3 (ii) Show that the subset {e,r,r², r³,r¹} is a normal subgroup.
Transcribed Image Text:6. The dihedral group D5 of isometries of a regular pentagon has elements {e,r,r², r³, r¹, x, rx, r²x, r³x, r²¹x} where r is a rotation by angle 27/5 and x, rx, r²x, r³x, ¹x are the five possible reflections. The multiplication table is determined by the fact that r has order 5, 2 has order 2 and ær = r¹²x. (i) Show by induction on n that r = rr for all n ≥0. (Note that r¹=r¹.) 3 (ii) Show that the subset {e,r,r², r³,r¹} is a normal subgroup.
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