# Introduction to Group Theory

## Cycles and Transpositions

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#### HOW TO PLAY

Now after all that reading, you can play with cycles and transpositions. For the cycle section of the page, click 'New Cycle' to have the computer give you a random permutation that can be expressed as 1 or more cycles. You get to find the cycles in the permutation. To input a cycle, type the cycle, in order, into the text box beside the 'Cycle In' button, with a comma following each number. Then press 'Cycle In'. if the cycle is correct, it will be printed in the text box below the permutation. If it's not correct you will be alerted.

After you've had fun playing with cycles, there is the transposition part. It's relatively easy. Just press the shuffle button, and start transposing. The goal is to match the bottom row with the top row. Input the positions of the suits that you want to switch in the two text boxes to the right of the 'Swap' button. Click it, and they'll change. Keep swapping the various suits until they match up.

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### Cycles

Your cycle, with values separated by commas:

 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

### Example 1: How to Play With Cycles

Pressing 'New Cycle' gives a random permutation of $$\left(\begin{array}{c}1&2&3&4&5&6&7&8&9\end{array}\right)$$. For this example, our permutation is $$\left(\begin{array}{c}4&5&2&9&3&6&8&7&1\end{array}\right)$$, so that the table above resembles: $$\left(\begin{array}{c}1&2&3&4&5&6&7&8&9\\4&5&2&9&3&6&8&7&1\end{array}\right)$$.

I will refer to the permuted, randomly generated, set as the "new" set and the origional set as the "old" set.

To start out, look at the old 1. It is above the new 4, so we can say that the 1 "turns into" or "goes to" the new 4. Now, find out where the old 4 "goes to". The old 4 is above the new 9, so the old 4 "goes to" the new 9. So far, we know that the cycle begins $$\left(\begin{array}{c}1&4&9&...\end{array}\right)$$, which is the same thing as (4 9 1) or (9 1 4). We only have a full cycle, however, when we come back to the first number in the cycle, which is a 1. Fortunately, old 9 goes to new 1 which is where we started from.

Type this complete cycle into the text field as "1,4,9" "4,9,1" or "9,1,4" and press "Cycle In". The cycle should show up in the box below the permutation.

Complete this process until you have fully decomposed the permutation into cycles, at which point you will be notified that you have succeeded. See if you can figure out the full cycle decomposition for this specific example (solution in next paragraph), and then try your hand out on some live, random ones.

The full cycle decomposition of this example is (remember, order does not matter, so your's might not exactly resemble this): $$\left(\begin{array}{c}1&4&9\end{array}\right)\left(\begin{array}{c}6\end{array}\right)\left(\begin{array}{c}7&8\end{array}\right)\left(\begin{array}{c}7&8\end{array}\right)\left(\begin{array}{c}2&5&3\end{array}\right)$$

### Transpositions

Rearrange the bottom to match the top.

 1 2 3 4

Pushing 'Shuffle' will give a new arrangement of suits. In order to arrange the bottom like the top, you have to select two of the positions (1 through 4) at a time. Continue swapping until you have successfully matched the bottom row with the top, at which point "congratulations" will appear and the font will turn green. Hit Shuffle to start over again.