MHB Prove that CnXCm isomorphic to Cgcd(m,n)XClcm(m,n)

[Alex224]

Hi, I have a problem in my homework that I am stuck with.
Let there be two natural numbers n,m.
let there be d = greatest common divisor of m and n - gcd(m,n)
and l = least common multiple of m and n - lcm(m,n)
I need to prove that CnXCm isomprphic to ClXCd (Cm Cn Cl Cd are all cyclic groups)

I have tried to see what happens if I look at m and n as products of prime numbers but I am kind of stuck around that idea without knowing where to take it.
I also think I should use the fact that gcd(m,n)*lcm(m,n) = m*n but also, can't figure out where to take it where to take it.
Another thing, if the gcd(m,n) = 1 I know that CmXCn is cyclic and the order of it is mn. I thought maybe this fact could help me somehow (m*n = l*d), but I don't know how, because it is possible that gcd(m,n) > 1

can somebody push me towards the right path to solution?
thanks!

 

[GJA]

Hi Alex224,

Are you still interested in a solution to this problem? I have something worked out and want to make sure you're still interested before doing a write up.

 

[Alex224]

Hi Alex224,

Are you still interested in a solution to this problem? I have something worked out and want to make sure you're still interested before doing a write up.

yes I am interested. thank you!

 

[GJA]

I'm not sure what book you're using or what theorems you my have come across in your studies, so I will quote the two we'll be needing here. Regardless, these are basic theorems for cyclic groups and will very likely be in any textbook you come across.

Theorem 1

If $G$ is a finite cyclic group with order $n$, then $G\cong \mathbb{Z}_{n}.$

Theorem 2

Let $n$ be a positive integer which has prime decomposition $n=p_{1}^{i_{1}}p_{2}^{i_{2}}\cdots p_{k}^{i_{k}}$, where $p_{1}<p_{2}<\ldots < p_{k}.$ Then $$\mathbb{Z}_{n}\cong\mathbb{Z}_{p_{1}^{i_{1}}}\times\mathbb{Z}_{p_{2}^{i_{2}}}\times\cdots\times\mathbb{Z}_{p_{k}^{i_{k}}}.$$

Exercise Proof Sketch

According to Theorem 1, it suffices to prove the exercise using $\mathbb{Z}_{m}$ and $\mathbb{Z}_{n}.$ Let $m$ and $n$ have the following prime decompositions: $m = p_{1}^{i_{1}}\cdots p_{k}^{i_{k}}$ and $n = q_{1}^{j_{1}}\cdots q_{s}^{j_{s}}.$

Case 1: $m$ and $n$ are relatively prime.

This is a short exercise using Theorem 2 above that I will let you try first.

Case 2: $m$ and $n$ are not relatively prime.

Since $m$ and $n$ are not relatively prime, they share prime factors in their decompositions. For the sake of concreteness, suppose they share $r$ common primes, where $1\leq r\leq \min\{k,s\}$. Since multiplication is commutative, we may assume without loss of generality that $p_{1} = q_{1},\ldots, p_{2} = q_{2},\ldots,$ and $p_{r} = q_{r}.$

Now, note that gcd$(m,n) = p_{1}^{*_{1}}\cdots p_{r}^{*_{r}},$ where $*_{t} = \min\{i_{t}, j_{t}\}$ for $1\leq t\leq r,$ and lcm$(m,n) = p_{1}^{**_{1}}\cdots p_{r}^{**_{r}}p_{r+1}^{i_{r+1}}\cdots p_{k}^{i_{k}}q_{r+1}^{j_{r+1}}\cdots q_{s}^{j_{s}},$ where $**_{t}=\max\{i_{t},j_{t}\}$ for $1\leq t\leq r.$

According to Theorem 2, it follows that

$$\mathbb{Z}_{m}\times\mathbb{Z}_{n}\cong \underbrace{\mathbb{Z}_{p_{1}^{i_{1}}}\times\cdots\mathbb{Z}_{p_{r}^{i_{r}}}\times\mathbb{Z}_{p_{r+1}^{i_{r+1}}}\times\cdots\times\mathbb{Z}_{p_{k}^{i_{k}}}}_{\mathbb{Z}_{m}}\times\underbrace{\mathbb{Z}_{p_{1}^{j_{1}}}\times\cdots\mathbb{Z}_{p_{r}^{j_{r}}}\times\mathbb{Z}_{q_{r+1}^{j_{r+1}}}\times\cdots\times\mathbb{Z}_{q_{s}^{j_{s}}}}_{\mathbb{Z}_{n}}\qquad (1).$$ From this point we are nearly done. I will leave it to you to try and use the facts about gcd$(m,n)$ and lcm$(m,n)$ above to rewrite equation (1) using the $p^{*}$ and $p^{**}$ terms. Once you have that, you can apply Theorem 2 above once more to obtain the result you want.

Please let me know if you have any further questions after making an honest attempt of your own. Good luck!