Question about center of mass of a system (human on ice)

[fluidistic]

Hello,
As I'm seeing the center of mass at university, many questions came to me but I could solve most of them. However I'm not sure about some points.
Say you have the system a human on ice. (without air. And consider the ice to be frictionless with the human). I could hardly understand how it would be impossible to the man to walk, but finally I understood. Even if he leaves off one of his legs perpendiculary to the ground, he won't be able to put it off upward perpendiculary leaving his other leg perpendicular to the ground. So he'll fall off onto the ground, leaving its center of mass exactly at the same point.
So I thought that future prisons could exist without cells but with a very well polished ground, until I realized that the man could exhale and thus moving where he wants. More than that, he could spit and move almost exactly where he wants... so it wouldn't work after all.

I've read that if there's no external force acting on the system and that the system is not moving then there's no way of how to change the position of the center of mass of the system. (if a man spits, he would "change" because he loses mass so it's not the same system).
But what if the man moves an arm horizontally? If his center of mass must not change of place, the man should move backward and then suddenly stop. But it's impossible since there's no friction between its foot and the ice. So is it possible to change the position of the center of mass with an internal force? (in this example the internal force is the muscular force that move the arm).
I hope you understand my question. If no, please tell me where I'm innacurate. Thanks!

 

[Doc Al]

If you are stationary on a frictionless surface, and you extend an arm out to the right, the rest of your body will be pushed to the left just enough to keep the center of mass right where it was. An internal force will not move the center of mass. Realize that whatever force your torso exerts on your arm, your arm exerts the same force in the opposite direction on your torso.

 

[fluidistic]

If you are stationary on a frictionless surface, and you extend an arm out to the right, the rest of your body will be pushed to the left just enough to keep the center of mass right where it was.

I don't understand how could it be like that. The rest of my body will be pushed to the left just enough to keep my center of mass where it was. That mean that my body will move and then stop to move, as my arm will complete its move. So my foot will slide on the ice and then stop to slide even if there's no friction? It seems that yes... how strange. But at least it explains that my center of mass cannot change its position. Ok!
Thank you once again Doc Al.