# Let σ a cycle of length s. Then σ^2 is a cycle iff s is odd.

1. Jun 20, 2012

### jmjlt88

Proposition: Suppose σ is a cycle of length s. Then σ2 is a cycle if and only if s is odd.

Quick remark... The following "proof" seems more like an explanation. This is for a self-study and I am looking back through my notes to refine/correct/improve. If you were correcting this, say out of 10pts, what would you give it?

Proof:

Let σ be a cycle of length s. That is, σ=(a1a2a3....as-1as). Suppose s is odd. Now, we calculate σ2 and we obtain that σ2=(a1a3a5a7....asa2a4a6....as-1). We notice that all odd numbered terms appear first in our cycle. Then, as, our last odd numbered term, gets sent to a2. The term a2 starts a sequence of moving even numbered terms to the next even numbered term, which concludes with as-1, which is also even. The term as-1 gets sent to a1 which completes the cycle. Now if s was even, then the term as-1 would be odd. But, as-1 would still get sent to a1. Hence, σ2 would be written as the product of disjoint cycles. That is,
[Equation 1] σ2=(a1a3a5a7....as-1)(a2a4a6....as). Thus, σ2 would not be a cycle.

Similarly, if we have that σ2 is cycle, then we deduce that s must be odd or else a situation similar to [Equation 1] would arise.

QED

Thanks for the help!! :)

2. Jun 21, 2012

### tiny-tim

hi jmjlt88!

sorry, but personally i'd give you 6, since it's a bit confusing and, although i think you've probably grasped the principles involved, i'm not convinced you have

since you apparently see how to make the cycles, it would be simpler just to show the single cycle and the pair of cycles, for the two cases

(also, i recommend using 2n and 2n+1, instead of the same symbol for both odd and even numbers )

3. Jun 21, 2012

### jmjlt88

Hey tiny-tim! Thanks! I actually did it like you said on paper. In my notes, I wrote the cycle in different notation [the two line notation where you put the elements on top and what they get sent to in the botton] which made it clearer. I then just observed what happened when s was odd, and then when s was even. Maybe if you saw that, I'd get a passing mark!! =)

4. Jun 22, 2012

### tiny-tim

hey jmjlt88!

(just got up :zzz:)

yes that looks better