# 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

1. If $X = [0,1]$, then $G$ is infinite.
2. If $X = [0,1]$, is $G$ abelian?

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I would suggest the following hints:

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

2. 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!

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(+1) indeed :) :) –  Ittay Weiss Apr 23 at 0:48

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?

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We have almost identical answers so +1 :) –  Amitesh Datta Apr 23 at 0:24