Is multiplicative mod Z9 cyclic? I'm asked if $\mathrm Z_9$or $\mathrm Z_{17}$ under multiplication is cyclic or not, proof needed. I understand the rules with addition, but under multiplication i can't make sens of the rules. I tried to manually calculate it with but did not find a generator thus i assumed that it is not cyclic but with $\mathrm Z_{17}$ it's a little bit more excruciating, what is the best way to prove if a group like this is cyclic or not? thanks in advance.

The multiplicative groups of $\mathbb{Z}/9 \mathbb{Z}$ and $\mathbb{Z}/ 17\mathbb{Z}$ are indeed cyclic.

More generally, the multiplicative group of $\mathbb{Z}/p^k \mathbb{Z}$ is cyclic for any _odd_ prime $p$.

If you are supposed to know this result, just invoke it. If you do not know this result, possibly you are expected to do this via a direct calculation.

To this end, you'd need to identify a generating element in each case. For example, for $9$ you have, trying $2$ as generator, $2^1= 2$, $2^2=4$, $2^2 =8=-1$, $2^4=-2$, $2^5=-4$, $2^6 = 1$.

Thus, $2$ indeed generates the multiplicative group of $\mathbb{Z}/9 \mathbb{Z}$, which has as its elements only the classes co-prime to $9$, that is, the six elements we got above.

However, the set $\mathbb{Z}/9 \mathbb{Z}$ (of nine elements) with multiplication, is not a group at all. For example, the class $0$ can never have a multiplicative inverse (neither have $3$ nor $6$).