Find all generators for the cyclic group $\bbmath Z_9 \times \bbmath Z_4?
I know I can use o(9,4)= 36 and all divisors of 36 are 1,2,3,4,6,9,12,18,36. This tells me that I will have subgroups of these orders.
2026-03-29 16:49:15.1774802955
Find all generators for the cyclic group $\mathbb Z_9 \times \mathbb Z_4$?
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Since $\phi(36)=12$, there are $12$ generators. There are several duplicates on how to find these:
How to find a generator of a cyclic group?
Finding all the generators of $Z_{n}$
Find all generators for the cyclic group $\mathbb{Z}_9 \times \mathbb{Z}_{10}$.
A useful fact is the following: if $g$ is a generator of the cyclic group $G$ of order $n$, then $g^{k}$ is a generator if and only if $\gcd(n, k) = 1$.