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Let $T$ be a rooted infinite binary tree and let $\text{Sym}(T)$ be the group of all symmetries of $T$.

  1. Show that any $\alpha \in \text{Sym}(T)$ sends the root to the root, even if you just view $T$ as an unrooted tree.

  2. Show that $\text{Sym}(T)$ contains an index 2 subgroup isomorphic to $\text{Sym}(T)\times\text{Sym}(T)$.

  3. Show that $\text{Sym}(T)$ is uncountable.

If anyone can offer any assistance on this problem, it would be greatly appreciated.

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