Algebra and Geometry | Part IA, 2002

(a) Find a subset TT of the Euclidean plane R2\mathbb{R}^{2} that is not fixed by any isometry (rigid motion) except the identity.

Let GG be a subgroup of the group of isometries of R2,T\mathbb{R}^{2}, T a subset of R2\mathbb{R}^{2} not fixed by any isometry except the identity, and let SS denote the union gGg(T)\bigcup_{g \in G} g(T). Does the group HH of isometries of SS contain GG ? Justify your answer.

(b) Find an example of such a GG and TT with HGH \neq G.

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