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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 $9$).

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    @Vafa: Perhaps you should verify that "the set of elements generated by $x$ (in $G$)" is **always** a subset of $G$ and **always** a subgroup of $G$, for any group $G$ and any element $x$.2011-04-02

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The set of elements generated by $1$ is necessarily a subgroup. In fact, if $\mathbb{Z}_9$ refers to the group of integers mod $9$ with addition, this subgroup is the entire group.

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    simply because $8=-1$ in ${\Bbb Z}_9$.2011-04-02
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How many times do you need to add eight to itself to get a multiple of nine? In other words, $9|x8 \Rightarrow ?|x$. The answer will be the order of the subgroup generated by 8.