Please see here Cyclic group test
Isn't this a mistake? Isn't this a cyclic group generated by $(5,1)$ ? thank you!
2026-03-25 20:35:05.1774470905
Is the following cyclic group test wrong in WolframAlpha?
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Yes! Wolfram alpha is wrong. It says also the group $\Bbb Z_3 \times \Bbb Z_2$ is not cyclic! , which is false.
The group $\Bbb Z_{13} \times \Bbb Z_{12}$ is cyclic of order $156$ since $\gcd\{13,12\}=1$ and $(5,1)$ is of order $ \text{lcm}\{13,12\}=156$, so $$\langle (5,1) \rangle=\Bbb Z_{13} \times \Bbb Z_{12}$$