Let G = D6 be the dihedral group of order 12, H be the subgroup of G generated by R120 where R120 is a counterclockwise rotation of 120°, and K be the subgroup of G generated by R180L where L is a reflection. Problem: a. Is H a normal subgroup of G? Justify your answer. b. Is HK = G? Justify your answer.

Let G = D6 be the dihedral group of order 12, H be the subgroup of G generated by R120 where R120 is a counterclockwise rotation of 120°, and K be the subgroup of G generated by R180L where L is a reflection. Problem: a. Is H a normal subgroup of G? Justify your answer. b. Is HK = G? Justify your answer.

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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Introduction of the problem:

Let G = D₆ be the dihedral group of order 12, H be the subgroup of G generated by R₁₂₀ where R₁₂₀ is a counterclockwise rotation of 120°, and K be the subgroup of G generated by R₁₈₀L where L is a reflection.

Problem:

a. Is H a normal subgroup of G? Justify your answer.

b. Is HK = G? Justify your answer.

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Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload it here. Introduction of the problem: Let G = D6 be the dihedral group of order 12, H be the subgroup of G generated by R120 rotation of 120°, and K be the subgroup of G generated by where R120 is a counterclockwise R180L where L is a reflection. Problem: What is (L R60)-1 in G?
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