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
These activities use permutations on 4 elements. They are
intended to be carried out after completion of Unit 1 on
Permutations in the Group Theory block. You will
 remind yourself about cycle notation
 practice composing permutations written in cycle notation
 experiment with transpositions and revise even and odd
permutations
 investigate subgroups of S_{4}
 explore conjugate permutations
Cycle notation activity
There are two standard notations for permutations, the two row
notation
and the cycle notation (1 2 4 3). Every permutation can be
written as a product of disjoint cycles. [Theorem 1 on p12 of the
Permutations unit.] In this activity you will compose permutations
written in cycle notation. The
ª © ¨ § symbols are to help
you keep track of the permutations. The applet will show the
effect of permutations on these symbols.
 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.
 Press `Reset'. Which element should you click on to show the permutation
 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.
 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.
 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.
 Were the answers the same? Press the 'Retrieve' button to see
them.
 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.
 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.
 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
Click on the link to `load applet on order' to load a new applet.
In this applet you can enter a permutation in cyclic notation and
find the powers of the permutation by pressing the `Apply' button.
The symbols on the left are to help you see the effect of the
permutations.
 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.
 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.
 What is the rule for the order of a permutation written in
cycle notation? [See Unit GT1 p19 Theorem 6].
 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
Click on the link to `load permutation applet' to reload the first
applet. In this activity you will show that all the permutations
in S_{4} can be written as products of transpositions, that is
cycles of length two. You will see which elements of S_{4} are
even permutations and which are odd permutations.
 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 '2cycles' in the drop down box that has 'All' at the top. this
will prevent cheating! How many transpositions did you use?
 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
(a_{1} a_{2} a_{3}¼a_{r}) = (a_{1} a_{r})(a_{1} a_{r1})¼(a_{1} a_{2}), 

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.
 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.)
 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 S_{4}
The aim of this activity is to reinforce ideas such as order of
an element, cyclic subgroup and more general subgroups.

 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.]
 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).
 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).
 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)?
 Use the applet to help you find the smallest subgroup
containing (1 2 4) and (1 4).
 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.
 Write down (1 2 3)^{1}, ((1 2)(3 4))^{1} and
(1 2 3 4)^{1}.
 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?
 Which element(s)k satisfy (1 2 3) = k(1 3 4)k^{1}?
What do you notice?
 Which element(s)k satisfy (1 2) = k(1 3)k^{1}?
What do you notice?
 Find all the conjugates of (1 2 3) in A_{4}, that is only
allowing conjugation by even permutations.
File translated from T_{E}X by T_{T}H, version 2.00.
On 01 Mar 2002, 15:07.