Express U(165) as an internal direct product of subgroups in four different ways.

Question

Express U(165) as an internal direct product of subgroups in four different ways.

I am trying to solve this problem. What I know is the following:

1. I understand how is Internal Direct Product related to External Direct Product.

2. I understand that U_k(n) is a subgroup of U(n) for each divisor k of n. In fact, it is the normal subgroup as U(n) is abelian.

The most similar question already asked here talks about stuff like Chinese Remainder Theorem which I don't want to delve into. Many of the solutions of this problem on the Internet are on subscription based websites and the partial solution that is displayed there doesn't talk about the Chinese Remainder Theorem.

Edited
More Context:
I am trying to solve this problem.
The closest I can find is this similar problem with example explanation but not proof.

As linked above, what is being done in the solution is that two coprime divisors k and t of 165 are found and U(165) is written as U_k($\frac{165}{k}$) . U_t($\frac{165}{t}$).

Answer 1

I'm unsure about how you might go about this without the Chinese Remainder Theorem, but I have an idea. Observe that $\vert U(165)\vert=\phi(165)=80=2^4\cdot5$. Now the first Sylow theorem implies that there is a $2$-Sylow subgroup $P$ of order $16$ and a $5$-Sylow subgroup $Q$ of order $5$. Furthermore, by the second Sylow theorem we know these are the unique subgroups of these orders since Sylow subgroups are conjugate and $U(165)$ is an abelian group. Now by observing that everything in $Q$ has order $5$ (other than the identity) and everything in $P$ has order $1,2,4,8,16$, we conclude that $P\cap Q=\{e\}$, and I will let you convince yourself that $PQ=QP=U(165)$. Thus, we have recognized $U(165)$ as an internal direct product of $P$ and $Q$. Now you wish to do this in four different ways. I don't know where to get the other three from here, but using the fact that these groups are abelian, you may consider using the structure theorem for abelian groups to figure out what the possibilities of $P$ can be, and then take subgroups of $P$ and extend $Q$ by things outside of the subgroup you took to get other inner direct products. I feel like this is much more heavy machinery to get to your answer than using the Chinese Remainder Theorem though.

Answer 2

By the CRT we have $$ U(165)\cong U(3)\times U(5) \times U(11)\cong C_2\times C_4\times C_{10}, $$ because $165=3\cdot 5\cdot 11$ and $U(p)\cong C_{p-1}$ for all prime numbers $p$. Of course we can also write $C_{10}\cong C_2\times C_5$ in this isomorphism. So we have three different possibilities: \begin{align*} U(165) & \cong C_2\times C_2\times C_4\times C_5,\\ & \cong C_2\times C_2\times C_{20},\\ & \cong C_2\times C_4\times C_{10}, \end{align*}

Reference:

Is it possible to prove that $U(p)$, for prime $p$, is cyclic using only group theory? If not, why not?