If I find the elements generated by $8$, can I say that the set of these elements is a subgroup of $\mathbf{Z}_9$ provided all these elements are in $\mathbf{Z}_9$? or to show that the set of these elements is a subgroup of $\mathbf{Z}_9$, should I test it for being a subgroup? if your answer is that no test is needed, then why is this so?
Edit: $\mathbf{Z}_9=\{0,1,2,3,4,5,6,7,8\}$ (the group under addition modulo $9$).