Determining if a factor group is cyclic.

Question

I am working with factor groups and have the following question.

Let $U(33)$ be the group of units in $\mathbb{Z_{33}}$. Let $\langle 4\rangle$ be the cyclic subgroup of $U(33)$. I want to determine whether or not $U(33)/\langle 4\rangle$ is cyclic.

One thing that I have stumbled upon is that:

$U(n)$ is cyclic if and only if $n=2,4,p^{\alpha},2 p^{\alpha}$.

Also,

If $G$ is cyclic then $G/H$ is cyclic.

Using the first result, I see that for $n=33 \neq 2,4,p^{\alpha},2 p^{\alpha}$ so I am tempted to say that $U(33)$ is not cyclic and hence $U(33)/\langle 4\rangle$ is not cyclic, however I am not very confident with that.

I am hoping that readers can offer a hint that will push me in the right direction. Also, if there are neater ways to go about such a problem I would be very interested to know.

Thank you.

Answer 1

Hint: The order of $4$ mod $33$ is $5$. The order of $U(33)$ is $20$. So, there are only two possibilities for the quotient.

Answer 2

$G/H$ can be cyclic without $G$ being cyclic; that is, the converse of the second theorem you quote is not true. So while $U(33)$ is not cyclic, this does not imply that the quotient you are interested in is not cyclic. However, the following hint might help you with your computation.

Hint: If $R$ and $S$ are rings, $(R\times S)^\times\cong R^\times\times S^\times$.

If you apply this result with $U(33) = (\Bbb Z/(33))^\times$ in mind (you'll have to figure out what the isomorphism in my hint is and what $4$ is mapped to via the isomorphism in the case of $U(33)$), you ought to be able to directly compute the quotient $U(33)/\langle 4\rangle$ without too much trouble.