No problem! Keep working at it and it will become second nature.

[radou]

Homework Statement

Let G be a group, and a an element of G of order m. What is the order of a^k?

The Attempt at a Solution

Well, first of all, if (a^k)^p = e, for some p, we have kp = mq, for some q. Now, for some k, the order of a^k is the least such p. Hence, it would make to sense to consider the least common multiple of k and m, lcm(m, n). The lcm(m, n)/k is the order of a^k. This is a bit informal perhaps, but it seems clear to me. Is this correct?

 

[radou]

Further on, it would follow from this that the powers of a which have the same order as a are those such that the least common multiple of k and m equals km.

 

[Dick]

Seems clear to me too. But you meant lcm(m,k), right? There's no 'n' in the problem.

 

[radou]

Seems clear to me too. But you meant lcm(m,k), right? There's no 'n' in the problem.

Oh, sorry, it was a typo - yes, I meant lcm(m, k). Thanks.

 

[Dick]

Further on, it would follow from this that the powers of a which have the same order as a are those such that the least common multiple of k and m equals km.

And if lcm(k,m)=km that tells you something about gcd(k,m).

 

[radou]

And if lcm(k,m)=km that tells you something about gcd(k,m).

Well, integers k, m such that lcm(k, m) = km tells us that k and m are relatively prime, hence gcd(k, m) = 1... I don't see a fact which follows from this right away.

 

[Dick]

Well, integers k, m such that lcm(k, m) = km tells us that k and m are relatively prime, hence gcd(k, m) = 1... I don't see a fact which follows from this right away.

Nothing you haven't already said. But the phrasing most people would use is that a^k generates the whole group if k and m are relatively prime.

 

[radou]

Nothing you haven't already said. But the phrasing most people would use is that a^k generates the whole group if k and m are relatively prime.

Yes, I'm aware of this fact - I proved it in another exercise. I'm still a bit new to group theory, so I don't realize some things right away yet.