i think that center of a body can not be outside of the body simply because its mass can not be outside of an object,it must be somewhere in the object.
I'm sure it can be. Just think about it. Try to explain: Get the same thing as the earth, drill a hole which is start in europa, going trough in the centre of the earth and come back on the other side on China. The gravity is always attract the things to the centre of the mass, which is geometrically the centre of the earth. But if the centre of the earth is empty, the gravity is going to attract the bodies with mass to the same place, so the centre of the mass, in other words to the empty place. If somebody has a question about this I can prove you this with a very simple experiment what anybody can do at home, and can see the truth.
I think Centre of Mass may be on the body or outside the body.
For example:- For a ring or a bangle the centre of mass must be in the middle part so as to get balanced. But the middle part is hollow. So its centre of mass is obtained by tieing two threads diagonally, which is outside the bangle.
CENTRE OF MASS AND CENTRE OF GRAVITY ARE TWO DIFFERENT CONCEPTS.
They may perhaps be different concepts in principle, but in practice the center of mass and
the center of gravity are always precisely the same point. And now that I think on it a bit
more, I believe they are one and the same concept as well.
The second paragraph above is spot-on treatment of the whole subject.