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Let S ⊆ R and let G be an arbitrary isometry of R . Prove that...

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that the symmetry group of G(S) is isomorphic to the symmetry group of S. Hint: If F is a symmetry of S, what is the corresponding symmetry of G(S)?

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Expert Solution

The Symmetry group of a set S is the group consisting of all bijections from S to itself with the group operation being function composition. An isometry is any map between R to R which preserves distances between points that is

|G(x) - G(y)|= |x-y|. Its immediate that an isometry is injective hence it is a bijection between its domain and its image.

Colby Messinger answered 1 year ago

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