[Abstract Algebra] Permutations and shuffling cards

1. Jun 23, 2013

Klungo

It's been a while since I've posted. This is a problem I had for a homework assignment a few weeks ago but I completely figure out. Any help appreciated.

1. The problem statement, all variables and given/known data
"A card-shuffling machine always rearranges cards in the same way relative to the order
in which they were given to it. All of the hearts arranged in order
from ace to king were put into the machine, and then the shuffled
cards were put into the machine again to be shuffled. If the cards
emerged in the order 10, 9, Q, 8, K, 3, 4, A, 5, J, 6, 2, 7, in what
order were the cards after the first shuffle?"

2. Relevant equations
\

3. The attempt at a solution

We have a permutation
$\alpha^2 = \left[ \begin{array}{cc} A & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & J & Q & K \\ 10 & 9 & Q & 8 & K & 3 & 4 & A & 5 & J & 6 & 2 & 7 \end{array} \right]$ which can be written as a single 13-cycle $\alpha^2 = (A,10,J,6,3,Q,2,9,5,K,7,4,8)$.

The goal is to find out how where each card is mapped to under $\alpha$.

2. Jun 23, 2013

haruspex

Think about raising α2 to various powers.

3. Jun 23, 2013

Klungo

From what I recalled on my first attempt.

We know that $(\alpha^2)^{13} = \alpha^{26} = e$ since $\alpha^2$ is a 13-single cycle. So, $|\alpha^{2}| \mbox{ divides } 13$.

That is, $|\alpha^2| = 1 \mbox{ or } 13$. Clearly, $|\alpha^2| \neq 1$ since $\alpha^2 \neq e$. Thus, $|\alpha^2| = 13$.

Now that I thought about it.

Using the formula $|g| = |g^k|gcd(k,|g|)$, we see that $|\alpha| = |\alpha^{2}| gcd(2,|\alpha|) = 13 gcd(2,|\alpha|) = 26 \mbox{ or } 13$.

Since we're "working" with the symmetric group $S_{13}$ and $26 = 2 \times 13$, we know that no element (by product of cycles) in $S_{13}$ has order 26. So, $|\alpha|=13$.

Hence, $\alpha = \alpha^{13} \alpha = \alpha^{14} = (\alpha^{2})^{7}$. Therefore, starting with A, we find $\alpha = (A,9,10,5,J,K,6,7,3,4,Q,8,2)$.

4. Jun 24, 2013

haruspex

Quite so - well done.

5. Oct 26, 2016

bsoblick

So does this imply that the order of the cards after the first shuffle were 9 A 4 Q J 7 3 2 10 5 K 8 6?

6. Oct 26, 2016

Yes.