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: What is the order of (R240 , R180L) in H⨁K ?

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: What is the order of (R240 , R180L) in H⨁K ?

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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Question

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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:

What is the order of (R₂₄₀ , R₁₈₀L) in H⨁K ?

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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 where R120 is a counterclockwise rotation of 120°, and K be the subgroup of G generated by R180L where L is a reflection. Problem: 1. Find an isomorphism from H to ℤ3