Calculating the order of an element in cycle notation

[Bashyboy]

Homework Statement

Let $g \in S_n$ and suppose that we know the cycle notation for $g$. How can one compute its order without repeatedly composing $g$ with itself?

Homework Equations

The Attempt at a Solution

[/B]
I took the group $S_4$ and tried formulating a conjecture. I composed several elements, such as (12), (13), (14), (1234), etc., and found that the order (the number of times one has to compose an element with itself to get the identity element) was equal to the length of the cycle minus one. I have been told that this is wrong, and have read elsewhere that the order is simply the length of the cycle.

Why is that so?

 

[Dick]

Homework Statement

Let $g \in S_n$ and suppose that we know the cycle notation for $g$. How can one compute its order without repeatedly composing $g$ with itself?

Homework Equations

The Attempt at a Solution

[/B]
I took the group $S_4$ and tried formulating a conjecture. I composed several elements, such as (12), (13), (14), (1234), etc., and found that the order (the number of times one has to compose an element with itself to get the identity element) was equal to the length of the cycle minus one. I have been told that this is wrong, and have read elsewhere that the order is simply the length of the cycle.

Why is that so?

(12) has length 2. (12)(12)=(12)^2=identity. What's the order of (12)?

 

[Bashyboy]

I figured it would be one, because you only have to compose (12) with itself once.

 

[Dick]

I figured it would be one, because you only have to compose (12) with itself once.

Composing a function with itself once is taking it to the second power.

 

[Bashyboy]

Oh, of course: the order is equal to the exponent.

 

[Bashyboy]

So, how would you go about proving something like this?

 

[Dick]

So, how would you go about proving something like this?

Calculate. Suppose you have a k cycle $\sigma=(n_1,n_2, ..., n_k )$. Then what is $\sigma(n_1)$? What is $\sigma^2(n_1)$? Etc.