Let $T$ be a rooted infinite binary tree and let $\text{Sym}(T)$ be the group of all symmetries of $T$.
Show that any $\alpha \in \text{Sym}(T)$ sends the root to the root, even if you just view $T$ as an unrooted tree.
Show that $\text{Sym}(T)$ contains an index 2 subgroup isomorphic to $\text{Sym}(T)\times\text{Sym}(T)$.
Show that $\text{Sym}(T)$ is uncountable.
If anyone can offer any assistance on this problem, it would be greatly appreciated.