1. ## order

If G is an abelian group of order pq, where p and q are relatively prime integers. Suppose G has an element v of order p and an element w of order q.
1)Prove that no powers of v can be equal to any power of w (except for e)
2)Prove the element vw has order pq.

2. Originally Posted by apple2009
If G is an abelian group of order pq, where p and q are relatively prime integers. Suppose G has an element v of order p and an element w of order q.
1)Prove that no powers of v can be equal to any power of w (except for e)
2)Prove the element vw has order pq.
For 1
Let v^x = w^y
v^(xp) = e = w^(yp)
=> q|yp => q|y (as gcd(p,q)=1)

Can you complete? 2) is a direct application of 1)

3. Originally Posted by aman_cc
For 1
Let v^x = w^y
v^(xp) = e = w^(yp)
=> q|yp => q|y (as gcd(p,q)=1)

Can you complete? 2) is a direct application of 1)
Aren't you missing the conclusion here?

4. Originally Posted by Drexel28
Aren't you missing the conclusion here?
Hi Drexel28 - I haven't exactly followed where I went wrong please. Would you mind elaborating a bit please?

5. Originally Posted by aman_cc
Hi Drexel28 - I haven't exactly followed where I went wrong please. Would you mind elaborating a bit please?
No. I don't think that you are wrong, it just lacks the conclusion.

Since if $v^x=w^y\implies q|y$ we may conclude that $w^y=e$. A similar argument shows that $v^x=e$ so they can only be equal when they are trivial. I agree with your solution it just didnt have the conclusion.

6. Originally Posted by Drexel28
No. I don't think that you are wrong, it just lacks the conclusion.

Since if $v^x=w^y\implies q|y$ we may conclude that $w^y=e$. A similar argument shows that $v^x=e$ so they can only be equal when they are trivial. I agree with your solution it just didnt have the conclusion.
o ok. Thanks. I intentionally left that for the OP to take it fwd.

7. Originally Posted by aman_cc
o ok. Thanks. I intentionally left that for the OP to take it fwd.
Then, my complete mistake. I apologize greatly for ruining your intended teaching method.