Question: why z is not a group?
Answer: The reason why (Z, *) is not a group is that most of the elements do not have inverses. Furthermore, addition is commutative, so (Z, +) is an abelian group. The order of (Z, +) is infinite. ... Note that 0 is an element of Zn and 0 is not coprime to any number so that is no inverse for 0.