Homeomorphism group I need to solve this problem, but I don't know how start.
Let be $(X,\tau)$ a topological space, and $G$ the set of all homeomorphism of $X$. I just proved that $G$ is a group, but I need also prove this


*

*If $X = [0,1]$, then $G$ is infinite.

*If $X = [0,1]$, is $G$ abelian?


I appreciate all your comments. Thank you.
 A: I would suggest the following hints:


*

*Consider the maps $x\to x^n$ for $x\in [0,1]$ and $n\in \mathbb{N}$.

*Consider the map $x\to 1-x$ for $x\in [0,1]$.
Exercise 1: Prove that there exists a homeomorphism of $[0,1]$ of infinite order.  (Note that this is stronger than 1.)
Exercise 2: Does there exist a homeomorphism of $[0,1]$ of order $k$ for each $k\in \mathbb{N}$?
Exercise 3: Prove that $G$ is uncountable.
Exercise 4: Does there exist a subgroup of $G$ isomorphic to $(\mathbb{R}, +)$?
Exercise 5: Can you exhibit a non-trivial proper normal subgroup of $G$?
Exercise 6: Prove that $G$ does not act transitively on $[0,1]$. What is the orbit of $\{x\}$ under the action of $G$ for $x\in [0,1]$?
In general, I recommend trying to come up with as many exercises as you can about $G$ in order to obtain a better understanding. 
Hope this helps!
A: 1) Start by constructing some homeomorphisms of the $[0,1]$. Hint: $x^2, x^3, x^4,\cdots $
2) Are there some homeomorphisms that reflect things?
