Thread: 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.

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)

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?

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?

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 we may conclude that . A similar argument shows that so they can only be equal when they are trivial. I agree with your solution it just didnt have the conclusion.

Originally Posted by Drexel28

No. I don't think that you are wrong, it just lacks the conclusion.

Since if we may conclude that . A similar argument shows that 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.

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.