hw7sol - Homework 7 1. Let P, be a rotation and let be any...

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Homework 7 1. Let ρ P,θ be a rotation and let β be any isometry. Prove that βρ P,θ β - 1 is a rotation. Solution: Let Q = β ( P ) so P = β - 1 ( Q ). Then we find βρ P,θ β - 1 ( Q ) = βρ P,θ ( P ) = β ( P ) = Q . So Q is a fixed point of βρ P,θ β - 1 . Since ρ P,θ is even, βρ P,θ β - 1 ( Q ) is also even. Now, βρ P,θ β - 1 ( Q ) is an even isometry which fixes the point Q , so it must be a rotation about Q (which might be the identity). 2. Let τ ~v be a translation and let β be any isometry. Prove that βτ ~v β - 1 is a translation (Hint: use 1.) Solution: The map βτ ~v β - 1 is an even isometry so it must be a rotation or translation.
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