Question

If we remove an electron from an element, does group number changes?

Answer 1

The element itself stays the same however. Only nuclear reactions can actually change the element itself by adding/removing protons and neutrons( neutron changes make isotopes which are still the same element though).

Answer 2

Removing a electron from a particle is not on will. It is due to the octet law.
So if you remove one electron like in same atoms in which it is possible due to their electrons in valence shell. They will either become positively charged or negatively charged.