Determine whether each of following is cyclic: $\Bbb Z_7, \Bbb Z_9, \Bbb Z_{10}, \Bbb Z_9^{\times}, \Bbb Z_{20}^{\times}$

Question

Determine whether each of the following is cyclic:

$$\Bbb Z_7, \Bbb Z_9, \Bbb Z_{10}, \Bbb Z_9^{\times}, \Bbb Z_{20}^{\times}.$$

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Theorems:

1) If $p$ and $q$ are primes then evry proper subgroup of order $p*q$ is cyclic.

2) $p\in \Bbb Z^{>0}$, $p$ is prime every and a group of order $p$ is cyclic and isomorphic to $\Bbb Z_p$.

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Attempt:

$\Bbb Z_7$ is cyclic by theorem 2. Not sure what generates it. $\Bbb Z_9^{\times}$ theorem 1 applies.

$ \Bbb Z_{20}$ and $ \Bbb Z_{20}^{\times}$ guessing no. Missing a theorem??

Should I use brute force and check the order of every element of $\Bbb Z_{20}$ because I will?

Answer 1

Every $ℤ_n$ is cyclic. For $ℤ_n^×$, there is a theorem that its elements are precisely elements of $ℤ_n$ coprime with $n$, so $\lvertℤ_n\rvert = φ(n)$. You can just write down the elements and observe the group structure.

Answer 2

$\mathbf Z_9^\times$ has order $\varphi(9)=6$ and it's abelian. There are only two isomorphism classes of groups of order $6$:

• a non-abelian group of order $6$ is isomorphic to the symmetric group $S_3$;
• an abelian group of order $6$ is isomorphic to the cyclic group $\mathbf Z/6\mathbf Z$.

For $\mathbf Z_{20}^\times$, as by the Chinese remainder theorem, $\;\mathbf Z /20\mathbf Z\simeq(\mathbf Z/4\mathbf Z)\times(\mathbf Z/5\mathbf Z)$, we have $$(\mathbf Z /20\mathbf Z)^\times\simeq(\mathbf Z/4\mathbf Z)^\times\times(\mathbf Z/5\mathbf Z)^\times\simeq\mathbf Z/2\mathbf Z\times \mathbf Z/4\mathbf Z,$$ which is not cyclic, as all its elements have order $\le 4$.