Let me quickly define my terms here:
A generating set for a group is a set of group elements that together can express every other element of the group (and no more) through products and inverses.
A basis or minimal generating set is a generating set such that if you remove any elements from the set, it is no longer a generating set.
I may have used nonstandard terminology, but this is how I think about it.
Anyways my question is:
Let G be a group
For any n less than |G|, does there exist a basis of this size n?
If anyone knows the answer to the analogous question with the group infinite, thats cool beans as well.
Thanks in advance guys.