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

[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...a_s-1a_s). Suppose s is odd. Now, we calculate σ² and we obtain that σ²=(a₁a₃a₅a₇...a_sa₂a₄a₆...a_s-1). We notice that all odd numbered terms appear first in our cycle. Then, a_s, 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 a_s-1, which is also even. The term a_s-1 gets sent to a₁ which completes the cycle. Now if s was even, then the term a_s-1 would be odd. But, a_s-1 would still get sent to a₁. Hence, σ² would be written as the product of disjoint cycles. That is,
[Equation 1] σ²=(a₁a₃a₅a₇...a_s-1)(a₂a₄a₆...a_s). 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! :)

 

[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 )

 

[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! =)

 

[tiny-tim]

hey jmjlt88!

(just got up :zzz:)

yes that looks better

except I'm worried about …

… I then just observed what happened when s was odd, and then when s was even.

… if by "observed" you mean explained in words, it would be clearer (and probably more convincing) if you just did it with maths, eg (2k 2(k+1) etc) and (2k+1 2(k+1)+1 etc)