-2

Dr. Pinter's "A Book of Abstract Algebra" presents the preface to a few exercises:

In $S_{5}$, express each of the following as the square of a cycle (that is, express $\alpha^{2}$ where $\alpha$ is a cycle).

I solved the first problem, $(132)$, by guessing and trying $(123)$.

In other words, Let $\alpha = (123)$. $\alpha^{2}$ = $(132)$.

However, I got stuck on the second problem. How can I figure out $\alpha$, such that $\alpha^{2}$ equals $(12345)$?

Kevin Meredith
  • 969

1 Answers1

2

You can try $(14253)$. Are you interested in a way of computing it?

mich95
  • 8,713
  • Is there a way to figure out $(14253)$ rather than simply trying it? If compute means an algorithm to figure out the square, then yes, please. – Kevin Meredith Jun 01 '15 at 19:30
  • Jup. First, you want to find such permutation $\alpha$. I will leave it to you to check that $\alpha$ must be a 5 cycle. Now, once that seen, we should start wrting the cycle. We start by (1**). The rest is to determine. Now $\alpha^{2}(1)=2$ so we must have $\alpha=(1*2)$. Now, we know $\alpha^{2}(2)=3$ so we must have $\alpha=(123)$. Can you figure it out ?(same method) – mich95 Jun 01 '15 at 19:32