A group is a set equipped with a binary operator that obeys a few rules. One of the rules is the existence of inverses. If the operator is addition, then your description is correct. That is, every element x of the underlying set has an additive inverse x^-1 also in the set such that x + x^-1 = 0. Obviously, zero needs to be in the set as well and in some constructions of the natural numbers it is not.
This is the graph theoretic definition of a tree.