Hello all. I reluctantly ask this question because it is probably,as the text states easy, but my desire to clear this point up overides my fear of looking a fool. I quote word for word but will use words instead of the belongs to symbol. A subset S of the set Z of integers is a subgroup of Z if 0 is in S, -x is in S,and x+y is in S for all x and y in S. It is easy to see that a non empty subset S of Z is a subgroup of Z if and only if x-y is in S for all x and y in S. I understand the definition and I can see that 0 is in S. I can only assume that somehow because -x ( the additive inverse of x ) is in S that this guarantees that x is in S. If S were a subgroup of Z of course -x being in the subgroup means that x was in it. But we are not assuming that S is a subgroup but testing for it. Help! Matheinste.