2

Let $\alpha$ and $\beta$ be permutation cycles of $\{1,2,\ldots,n\}$ such that $\alpha^2=\beta^2$ Can we conclude that $\alpha=\beta$, if

(a) $\alpha,\beta$ are odd?

(b) $\alpha,\beta$ are even?

We can write $\alpha=(a_1 a_2\ldots a_k), \beta=(b_1 b_2\ldots b_l)$, and we know that $\alpha^2=\beta^2$. I don't know how to continue to get more information on $\alpha,\beta$.

Community
  • 1
Kunal
  • 2,739

1 Answers1

3

Note that a cycle is odd if it has even length, and conversely.

Just experiment a little, and you will see that (a) is false.

For (b), notice that if $\alpha$ is a cycle of odd length, then $\alpha^2$ is also a cycle, of the same length as $\alpha$. (This is not true if $\alpha$ has even length.) Can you recover $\alpha$ from $\alpha^2$ ?

fkraiem
  • 3,129