Multiplying non-disjoint permutation cycles

[Saint Medici]

Maybe I'm just being dense, but I've been having issues with the multiplying non-disjoint permutation cycles (as you may have guessed from the topic title). Simple products like (1, 4, 5, 6)(2, 1, 5) [an example from my textbook], as well as in the opposite order. Mayhap that I'm tired.

Disjoint cycles are quite easy, so I'm not sure what's going on in my head that fails me so.

-x

 

[StatusX]

Just keep track of each element, keeping in mind that the order is reversed, eg, ab is the permutation b followed by the permutation a. So, in your example, 1 is sent by the second cycle (in the order they're written) to 5, then 5 is sent by the first to 6, so 1 ends up at 6. 6 is unmoved by the second cycle, then moved by the first to 1. Thus one of the cycles in the product is the transposition (1 6) (usually these are written without commas).

 

[Edgardo]

Here are some links with nice tutorials:

1) Permutation groups in the blog "The unapologetic mathematician" here
This is where I first learned about the cycle notation.

2) Tom Davis http://mathcircle.berkeley.edu/BMC3/perm/perm.html contains a chapter on the
http://mathcircle.berkeley.edu/BMC3/perm/node3.html . Read the next pages too.

3) Short article at Wolfram Mathworld

4) Sarah Kitchen's notes on Permutation groups and polynomials. Read the chapter on "Cycle Decomposition" on page 2 and 3.

5) Jaap's puzzle explains permutations with diagrams

 

[Edgardo]

I just found an awesome interactive flash video on permutations.
First of all watch the graphics http://nrich.maths.org/public/viewer.php?obj_id=2781&part=index.

Then watch the tutorials on how to use the interactive flash video:
http://nrich.maths.org/public/viewer.php?obj_id=2789&part=. Watch the first tutorial "Making Shuffles"
and then try out producing some shuffles with the
http://nrich.maths.org/public/viewer.php?obj_id=2741&part=index. Then watch the second tutorial "Manipulating Shuffles".
You should then be able to calculate
your example (1 4 5 6)(2 1 5) with the Shuffle Factory Program.

(i) Just start by creating the (1 5 2) shuffle and the (1 4 5 6) shuffle.
Note that (2 1 5) is the same as (1 5 2).
Of course, both shuffles have to consist of "6 pearls".
Make a copy of both and put them on the right side.

(ii) Now work with the copies:
Put the (1 4 5 6) below the (2 1 5) shuffle like this:

(1 4 5 6)
(2 1 5)

Then merge them together and see the result by first clicking on the PLAY button
in the right bottom corner of the merged shuffle. You can follow the path of the pearls.
Then press the button in the left bottom corner of the shuffle to finally merge the two shuffles together.
See the result.

This is really fun!