Here is the problem, I thought it the I've been thinking for a long time, but I still don't have any ideas.
Problem: Let A(S) be the group of all bijective functions on S with composition as its binary operation. then A(S) is finitely generated if and only if S is finite set.
I know that proving from the right to the left is relatively easier, but when it comes to proving from the left to the right, I really don't have any ideas. I hope someone can help.
Assuming that $S$ is infinite, $A(S)$ is uncountable, but if it were finitely generated, it would be countable. So, if $A(S)$ is finitely generated, $S$ must be finite. (Added later: Going in the other direction, if $S$ is finite, $A(S)$ is finite, so it's finitely generated.)