Activity on Permutations

This is an interactive web page containing an applet. It should load and run in any reasonably modern browser. Solutions to questions can be viewed by clicking a button. You may, if you wish download the pdf file of the solutions. Similarly you may, if you wish, download the pdf file of the activity. These will need Adobe acrobat to read and print.

Proposed Learning Outcomes

• remind yourself about cycle notation
• practice composing permutations written in cycle notation
• experiment with transpositions and revise even and odd permutations
• investigate subgroups of S₄
• explore conjugate permutations

Cycle notation activity

1. Click on the element (1 2 3). The symbols ¨ ª © § should appear. Write down the permutation (1 2 3) in two row notation. Press 'Compose' before you read the solution.
2. Press `Reset'. Which element should you click on to show the permutation

   æ ç è 1 2 3 4 2 1 4 3 ö ÷ ø ?
3. Press `Reset'. Click on (1 2). Then click on (3 4). Notice that the two permutations appear on the right of the applet with the one you clicked on first to the right. This is the way that you write the composition of the two permutations. As they are functions, it is the one on the right that is applied first. What single permutation has the same effect as these two in succession? Press the `Compose' button to see if you were right.

4. Press `Reset'. Click on (1 3). Then click on (1 2). Write down the single permutation that has the same effect as these two in succession? Press the `Compose' button to see if you were right. Press the 'Store' button.
5. Press `Reset'. Click on (1 2). Then click on (1 3). Write down the single permutation that has the same effect as these two in succession? Press the `Compose' button to see if you were right. Press the 'Store' button.
6. Were the answers the same? Press the 'Retrieve' button to see them.
7. Press `Reset'. Click on (1 2 3) and then on (1 2). Write down the composition of these two permutations in cycle notation. Press 'Compose' to check your answer.
8. Press `Reset'. Click on (2 3 4), then (1 2), and then (1 4 2 3). Write down the composition of these three permutations in cycle notation. Press 'Compose' to check your answer.
9. Press `Reset'. Practice with various numbers of permutations until you feel confident that you can compose permutations in cycle notation. (You can compose up to 9 permutations.)

Activity on order of permutations

1. Click on (1 2 3)(5 6) at the top of the applet. What are the lengths of the cycles in this permutation? What order do you think this permutation has? Click on `Apply' repeatedly until the identity permutation, (1)(2)(3)(4)(5)(6) appears. How many times did you click? The answer is in the box at top right of the applet. This is the order of the permutation.
2. Press 'Clear'. Click on (1 2 3 4)(5 6), which is second in the drop down box at the top of the applet. What are the lengths of the cycles in this permutation? What order do you think this permutation has? Click on `Apply' repeatedly until the identity permutation appears. How many times did you click? The answer is in the box at top right of the applet. This is the order of the permutation.
3. What is the rule for the order of a permutation written in cycle notation? [See Unit GT1 p19 Theorem 6].

4. Write down the the order of (1 2 3 4)(3 6)? Press 'Clear'. Click on 'User' in the drop down box and enter (1234)(36). Include the brackets but do not include any spaces. Press 'Apply'. What happens? What is the order of the permutation?

Activity on transpositions

1. Press `Reset'. Click on the permutation (1 2 3 4). Press `Target'. Notice that the effect of the permutation on the symbols ª,©,¨,§ has appeared at the top right of the applet and everything else has been reset. Your task is to reach the target using just the transpositions. Click on '2-cycles' in the drop down box that has 'All' at the top. this will prevent cheating! How many transpositions did you use?
2. Press `Reset'. Now make (1 2 3) the target. How many permutations do you use this time?

   You may have used the strategy given on p20 of the GT1 unit on Permutations to do the last two questions. The strategy which gives

   (a1 a2 a3¼ar) = (a1 ar)(a1 ar-1)¼(a1 a2),

   tells us that a cycle of even length requires an odd number of transpositions and a cycle of odd length requires an even number of transpositions. The following activity requires a different strategy.
3. Press `Reset'. An adjacent transposition is a two cycle consisting of consecutive numbers, such as (2 3). Repeat the previous two exercises (for (1 2 3 4) and (1 2 3)) using just adjacent transpositions. (Click on 'Adjacent' in the drop down box.) ( Note that the number of transpositions used remains respectively odd and even.)
4. Press 'Reset'. Find the composition of (1 3 2)(1 2 4 3). Make this the target. Find two different ways of reaching this target using adjacent transpositions.

Subgroups of S₄

1. • Press 'Reset'.
   • Click on the element (1 2).
   • Press 'Store'.
   • Click again on the element (1 2).
   • Press 'Compose' and press 'Store'.
   • Now press 'Retrieve'.

   A window will appear showing the elements you have stored - in this case (1 2) and e. You generated these elements by repeated application of (1 2). Since you have generated the identity, there are no more elements obtainable by applying (1 2) again. The cyclic subgroup generated by (1 2) consists of these elements. Since there are two elements in the list (1 2) has order 2. Close the window and press 'Clear Store'.

   [Note: resetting will close the window and clear the permutations. It will not clear the list of elements stored.]

2. Store the elements generated by (1 2 3). Remember to press 'Reset' after you have used 'Compose'. What is the order of (1 2 3)? What are the elements of the cyclic subgroup generated by (1 2 3).

3. Press 'Clear Store'. Store the elements generated by (1 2 3 4). What is the order of (1 2 3 4)? What are the elements of the cyclic subgroup generated by (1 2 3 4).

4. What is the order of (1 3)(2 4). What are the orders of the elements of the cyclic group generated by (1 2 3 4)?
5. Use the applet to help you find the smallest subgroup containing (1 2 4) and (1 4).

6. Use the applet, if necessary, to help you find the smallest subgroup containing (1 3) and (2 4).

Conjugacy

We say that the group element g is conjugate to h in the group G if there is an element k Î G satisfying g = khk^-1. In this section you will explore some of the properties of conjugate permutations. It revises and consolidates the material in the tape sections from pages 28 to 31. You are advised to work through this before doing this activity. You should use strategies given in the tape section, where appropriate, and use the applet to confirm your answers.

1. Write down (1 2 3)^-1, ((1 2)(3 4))^-1 and (1 2 3 4)^-1.

2. Use the applet to find (1 2 3)(1 2 3 4)(1 2 3)^-1. What is the cycle form of the result? What about other conjugates of (1 2 3 4)? Now try conjugates of (1 2)(3 4). What do you notice?

3. Which element(s)k satisfy (1 2 3) = k(1 3 4)k^-1? What do you notice?

4. Which element(s)k satisfy (1 2) = k(1 3)k^-1? What do you notice?

5. Find all the conjugates of (1 2 3) in A₄, that is only allowing conjugation by even permutations.