I've looked in my abstract algebra text and just remembered that the group axioms may be stated in alternative forms. So my text says:

1. The set must be closed under the binary operation

2. For all members of the set associativty of the binary operation must hold

3. There must be identity el3ment

4. !! If A is in the group, there must be element A' in the same group such that both (!!) AA'=e and A'A=e hold

ugh....