1. element

Suppose $\displaystyle a,b \in G$ and $\displaystyle ab = ba$. If $\displaystyle |a|$ and $\displaystyle |b|$ are relatively prime, prove that $\displaystyle ab$ has order $\displaystyle |a||b|$.

So suppose $\displaystyle a^m = e$ and $\displaystyle b^n = e$. Then $\displaystyle \gcd(m,n) = 1$. Thus $\displaystyle 1 = \frac{\text{lcm}(m,n)}{mn}$ or $\displaystyle mn = \text{lcm}(m,n)$. So $\displaystyle (ab)^{mn} = (a^{m})^{n} \cdot (b^{n})^{m} = e^{n}e^{m} = e$. Thus the order of $\displaystyle ab$ is $\displaystyle |a||b|$.

Is this correct?

2. Originally Posted by Sampras
Suppose $\displaystyle a,b \in G$ and $\displaystyle ab = ba$. If $\displaystyle |a|$ and $\displaystyle |b|$ are relatively prime, prove that $\displaystyle ab$ has order $\displaystyle |a||b|$.

So suppose $\displaystyle a^m = e$ and $\displaystyle b^n = e$. Then $\displaystyle \gcd(m,n) = 1$. Thus $\displaystyle 1 = \frac{\text{lcm}(m,n)}{mn}$ or $\displaystyle mn = \text{lcm}(m,n)$. So $\displaystyle (ab)^{mn} = (a^{m})^{n} \cdot (b^{n})^{m} = e^{n}e^{m} = e$. Thus the order of $\displaystyle ab$ is $\displaystyle |a||b|$.

Is this correct?
I suppose that is fine. Are you allowed to use that result? It makes the proof kind of trivial.

3. Originally Posted by Drexel28
I suppose that is fine. Are you allowed to use that result? It makes the proof kind of trivial.
What would you suggest?

4. Originally Posted by Sampras
Suppose $\displaystyle a,b \in G$ and $\displaystyle ab = ba$. If $\displaystyle |a|$ and $\displaystyle |b|$ are relatively prime, prove that $\displaystyle ab$ has order $\displaystyle |a||b|$.

So suppose $\displaystyle a^m = e$ and $\displaystyle b^n = e$. Then $\displaystyle \gcd(m,n) = 1$. Thus $\displaystyle 1 = \frac{\text{lcm}(m,n)}{mn}$ or $\displaystyle mn = \text{lcm}(m,n)$. So $\displaystyle (ab)^{mn} = (a^{m})^{n} \cdot (b^{n})^{m} = e^{n}e^{m} = e$. Thus the order of $\displaystyle ab$ is $\displaystyle |a||b|$.

Is this correct?

I'm afraid it's not: you only proved that $\displaystyle (ab)^{mn}=e$ which is pretty trivial. The interesting part is in showing that there is not power of ab less than mn which equals the unit...

Tonio

5. Originally Posted by tonio
I'm afraid it's not: you only proved that $\displaystyle (ab)^{mn}=e$ which is pretty trivial. The interesting part is in showing that there is not power of ab less than mn which equals the unit...

Tonio
You have to use fact that ab=ba, because this isn't true if this condition is not met? So $\displaystyle (ab)^{mn} = (ba)^{mn}$. Suppose $\displaystyle (ab)^{mn- \varepsilon} = e = (ba)^{mn- \varepsilon}$. Then $\displaystyle \frac{(ab)^{mn}}{(ab)^{\varepsilon}} = e = \frac{(ba)^{mn}}{(ba)^{\varepsilon}}$ for some $\displaystyle \varepsilon \in \mathbb{Z}$.

Or $\displaystyle \frac{a^{mn}b^{mn}}{a^{\varepsilon}b^{\varepsilon} } = e$. Or $\displaystyle \frac{e}{a^{\varepsilon} b^{\varepsilon}} = e$ so that $\displaystyle (ab)^{\varepsilon} = e$. Contradiction?

6. NOTE: I am actually not sure if this is correct. Let another member validate this. I feel as though I may have made a stupid incorrect assumption.

Problem: Let $\displaystyle (m,n)=1$. Prove that if $\displaystyle |a|=m,|b|=n$ that $\displaystyle |ab|=mn$.

Proof: Clearly $\displaystyle \left(ab\right)^{mn}=e$. Now suppose that $\displaystyle |ab|=\omega<mn$ then $\displaystyle (ab)^{\omega}=e\implies a^{\omega}=b^{-\omega}$ and $\displaystyle b^{\omega}=a^{-\omega}$. Thus $\displaystyle \left(b^{-\omega}\right)^m=\left(a^{\omega}\right)^m=e\impli es a^{\omega\cdot m-\omega\cdot n}=e$. Similarly $\displaystyle b^{\omega\cdot n-\omega\cdot m}=e$. And since $\displaystyle |a|=m$ we see that $\displaystyle m|\omega\cdot m-\omega\cdot n\implies m|\omega n$ and since $\displaystyle (m,n)=1$ this is only true if $\displaystyle m|\omega$. Similarly $\displaystyle n|\omega\cdot n-\omega\cdot m$ and since $\displaystyle |b|=n$ this means $\displaystyle n|\omega\cdot n-\omega\cdot m\implies n|\omega\cdot m\$ and by previous reasoning $\displaystyle n|\omega$. Therefore $\displaystyle m,n|\omega$ which means that $\displaystyle \text{lcm}(m,n)|\omega$ but since $\displaystyle (m,n)=1$ we have that $\displaystyle \text{lcm}(m,n)=mn$ and $\displaystyle mn|\omega$ which is a contradiction since $\displaystyle 0<\omega<mn$.

7. Originally Posted by Drexel28
NOTE: I am actually not sure if this is correct. Let another member validate this. I feel as though I may have made a stupid incorrect assumption.

Problem: Let $\displaystyle (m,n)=1$. Prove that if $\displaystyle |a|=m,|b|=n$ that $\displaystyle |ab|=mn$.

Proof: Clearly $\displaystyle \left(ab\right)^{mn}=e$. Now suppose that $\displaystyle |ab|=\omega<mn$ then $\displaystyle (ab)^{\omega}=e\implies a^{\omega}=b^{-\omega}$ and $\displaystyle b^{\omega}=a^{-\omega}$. Thus $\displaystyle \left(b^{-\omega}\right)^m=\left(a^{\omega}\right)^m=e\impli es a^{\omega\cdot m-\omega\cdot n}=e$. Similarly $\displaystyle b^{\omega\cdot n-\omega\cdot m}=e$. And since $\displaystyle |a|=m$ we see that $\displaystyle m|\omega\cdot m-\omega\cdot n\implies m|\omega n$ and since $\displaystyle (m,n)=1$ this is only true if $\displaystyle m|\omega$. Similarly $\displaystyle n|\omega\cdot n-\omega\cdot m$ and since $\displaystyle |b|=n$ this means $\displaystyle n|\omega\cdot n-\omega\cdot m\implies n|\omega\cdot m\$ and by previous reasoning $\displaystyle n|\omega$. Therefore $\displaystyle m,n|\omega$ which means that $\displaystyle \text{lcm}(m,n)|\omega$ but since $\displaystyle (m,n)=1$ we have that $\displaystyle \text{lcm}(m,n)=mn$ and $\displaystyle mn|\omega$ which is a contradiction since $\displaystyle 0<\omega<mn$.
Suppose $\displaystyle a^{r}b^{r} = e$ for some $\displaystyle r \in \mathbb{Z}^{+}$. Then $\displaystyle b^{nr-r}b^{r} = b^{nr} = e$. Let $\displaystyle t = nr-r$. Then $\displaystyle a^{r} = b^t$. So $\displaystyle (a^r)^n = (b^t)^n$ so that $\displaystyle a^{rn} = e$. So $\displaystyle m$ divides $\displaystyle rn$ but $\displaystyle (m,n) = 1$. Thus $\displaystyle m|r$ so that $\displaystyle m$ divides $\displaystyle |a|b|$. Similarly, $\displaystyle n$ divides $\displaystyle |a|b|$.

8. Originally Posted by Sampras
Suppose $\displaystyle a^{r}b^{r} = e$ for some $\displaystyle r \in \mathbb{Z}^{+}$. Then $\displaystyle b^{nr-r}b^{r} = b^{nr} = e$. Let $\displaystyle t = nr-r$. Then $\displaystyle a^{r} = b^t$. So $\displaystyle (a^r)^n = (b^t)^n$ so that $\displaystyle a^{rn} = e$. So $\displaystyle m$ divides $\displaystyle rn$ but $\displaystyle (m,n) = 1$. Thus $\displaystyle m|r$ so that $\displaystyle m|ab$. Similarly $\displaystyle n|ab$.
So wait haha. Are you agreeing or disagreeing with me?

9. Originally Posted by Drexel28
So wait haha. Are you agreeing or disagreeing with me?
It's essentially the same proof. Except just choosing an arbitrary r to begin with.

10. Originally Posted by Sampras
It's essentially the same proof. Except just choosing an arbitrary r to begin with.
Ok. Good. Hope that helped.