Your definition of a group is incorrect.
The line
(3) There is an element 0 in S such that a op 0 = 0 for all a in S
should be
(3) There is an element 0 in S such that a op 0 = a for all a in S
The set {-1, 0, +1} does not form a group under the operation of multiplication.
The number +1 does act as an identity element for this set, but the number 0 has no inverse, which is required by
(4) If a in S, then there is a b in S such that a op b = b op a = 0
The set {-1, +1} does form a group under multiplication.