1.II.5B

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

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

(b) Find an example of such a $G$ and $T$ with $H \neq G$.

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