Checking if a quotient group is cyclic

Question

Is there an easy way to check if the group $(\mathbb{Z}_9 \times \mathbb{Z}_9)/\langle(3,3)\rangle$ is cyclic? I know I could determine this by writing out each of the elements and checking if one of them is a generator. But there are 27 elements (if my math is correct), which would make that approach very tedious. Is there an easier way?

Answer 1

It isn't. In the quotient, the order of any element's image under the canonical projection $\pi$ divides its order in $\Bbb Z_9\times\Bbb Z_9$. Thus it divides $9$.

Answer 2

The map $\varphi\colon \Bbb Z_9\times \Bbb Z_9\to \Bbb Z_3\times \Bbb Z_9$, defined by $([m]_9,[n]_9)\mapsto ([m]_3,[m-n]_9)$, is a surjective homomorphism with kernel $\langle(3,3)\rangle$. By the First Homomorphism Theorem, $(\Bbb Z_9\times \Bbb Z_9)/\langle(3,3)\rangle\cong \Bbb Z_3\times \Bbb Z_9$, which is not cyclic, for $\operatorname{gcd}(3,9)=3\ne 1$.