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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.

cele
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  • Since $\gcd(9,4)=1$, $(x,y)$ is a generator if and only if $x$ generates $\mathbb Z_9$ and $y$ generates $\mathbb Z_4$. – Milten Nov 04 '19 at 11:49
  • More closely a duplicate of https://math.stackexchange.com/q/1512377/620957 – Milten Nov 04 '19 at 12:07

1 Answers1

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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 $\gcd(n, k) = 1$.

Dietrich Burde
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