The direct product of two cyclic groups of coprime order is another cyclic group
= The direct product of two cyclic groups of coprime order is another cyclic group
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You just map the value (1, 1) $C_m \times C_n$ to the value 1 of $C_{mn}$, and it works out. E.g. for $C_2 \times C_3$, the <generating set of a group>[group generated by] of (1, 1) is:
``
0 = (0, 0)
1 = (1, 1)
2 = (0, 2)
3 = (1, 0)
4 = (0, 1)
5 = (1, 2)
6 = (0, 0) = 0
``