Permutation Groups:

Permutation Groups: Cycles, Transpositions, and FUTURAMA Bob Gardner KME Advisor Fall 2013

Futurama is an animated series about a pizza delivery boy, Philip J. Fry, who time travels from New Year’s Eve 1999 to New Year’s Eve 2999. The show follows the underachieving Fry as he deals with talking robots, one eyed mutants, and various space aliens. The show aired from March 28, 1999 to September 4, 2013 on the Fox network (1999 to 2003) and Comedy Central (2008 to 2013),for a total of 140 shows. The show was created by Matt Groening, of The Simpsons fame.

Futurama has always been big on math and science related jokes. In fact, several of the writers have undergraduate and graduate degrees in mathematics, computer science, physics, and chemistry. Dr. Sarah Greenwald of Appalachian State University has a webpage devoted to the mathematical content of Futurama:http://mathsci.appstate.edu/~sjg/futurama/

Writer Ken Keeler has a Ph.D. in applied math from Harvard University. He wrote the script for “The Prisoner of Benda” which first aired on Comedy Central on August 19, 2010. The plot concerns a body switching machine which allows two bodies to switch brains, but once a pair of bodies has switched brains then the same pair of bodies cannot switch again (due to the cerebral immune response). After several uses of the machine, the problem is to return the brains to the correct bodies. Keeler proved a theorem about permutation groups and it is used in the show.

The Regular Characters Fry (F) Leela (L) Zoiberg (Z) Bender (B) Amy (A) Hermes (H) Professor Farnsworth (P)

Characters Relevant to “The Prisoner of Benda” Emperor Nikolai (E) Sweet Clyde (S) Washbucket (W) Bubblegum Tate (T)

Groups • Permutation Notation • Permutation Group • Transpositions • The Inversion Theorem

Definition. A binary operation on a set is a rule that assigns to each ordered pair of elements of the set an element of the set. The element associated with ordered pair (a,b) is denoted Example. Consider the set M2 of all matrices. A binary operation on M2 is matrix multiplication. Notice that this binary operation is not commutative. Leela (L)

Definition. A group is a set G together with a binary operation on G such that: • The binary operation is associative • There is an element e in G such that for all , and • For each there is an element with the property that Note. Element e is called the identity element of the group and element is called the inverse of element a. Amy (A)

Example. Here is an example of a “multiplication table” of a group with binary operation and set * G={0,1,2,3,4}. G={a,b,c,d,e}. Professor Farnsworth (P) The integers modulo 5, Z5.

Example. In general, the integers modulo n form a group under addition, Zn. Example. The integers form an infinite group under addition, Z. The integers do not form a group under multiplication or division. Example. The real numbers form a group under addition, R. Zoiberg (Z)

Definition. A permutation of a set A is a one-to-one and onto function from set A to itself. Example. If A={1,2,3,4,5}, then a permutation is function s where: s(1)=4, s(2)=2, s(3)=5, s(4)=3, s(5)=1. This can be represented with permutation notation as: Bender (B)

Example. Another permutation on set A={1,2,3,4,5} is We can multiply two permutations to get the permutation product as follows: Fry (F)

Example. We can write the previous permutation as a “product of cycles”: Theorem. Every permutation of a finite set is a product of disjoint cycles. Hermes (H)

Example. When taking products of cycles, we read from left to right: 1 4 5 6 2 1 5 4 1 3 2 6 5 Amy (A)

Example. Consider the two permutations on {1,2,3}: and The product of these permutations is: For this reason, is called the inverse of Bender (B)

Note. In fact, the permutations of a set form a group where the binary operation is permutation product. Definition. The group of all permutations of a set of size n is called the symmetric group on n letters, denoted Sn. Note. Two important subgroups of Snare the dihedral groupDn and the alternating groupAn.

Definition. A cycle of length two is a transposition. Theorem. Any permutation of a finite set of at least two elements is a product of transpositions.

Now let’s watch “The Prisoner of Benda.” Watch for the seven main characters (Fry, Leela, Amy, Professor Farnsworth, Zoiberg, Hermes, and Bender), along with the other four characters: Washbucket (W) Sweet Clyde (S) Emperor Nikolai (E) Bubblegum Tate (T)

Note. We will represent a body switch with a transposition. The first body switch is between the Professor and Amy. We denote this as: or

After the First Three Body Switches… …the permutation is

Let’s explore some permutations and their inverses…

Suppose we start with this population… YELLOW GREEN RED BLUE

...and permute the red and blue disks: YELLOW GREEN RED BLUE

We can… YELLOW GREEN (Y,R) (G,B) (Y,B) (G,R) RED BLUE

Finally, we interchange Y and G YELLOW GREEN (Y,G) RED BLUE

Suppose we start with this population… YELLOW GREEN RED BLUE BLACK

Let’s mix things up to create a 3-cycle: YELLOW GREEN (B,Bc) (R,Bc) RED BLUE BLACK (RED,BLUE,BLACK)

Let’s mix things up to create a 3-cycle: YELLOW GREEN (G,Bc) (Y,B) (G,R) (Y,G) (Y,R) (G,B) RED BLUE BLACK

In The Prisoner of Benda, the initial transpositions are: This yields the resulting permutation:

Note. The problem is to find an inverse of the permutation which does not repeat any of the previous transpositions (or repeat any new transpositions). Notice that this transposition can be written as the disjoint cycles:

Futurama gives the following transpositions which produce the inverse of the above permutation:

The transpositions give the permutation: which equals the product of disjoint cycles:

We can take the permutation product to get: This verifies that Futurama’s second permutation is the inverse of the first permutation.

In terms of cycles, we have:

Sweet Clyde’s Inversion Theorem. Any permutation on a set of n individuals created in the body switching machine can be restored by introducing at most two extra individuals and adhering to the cerebral immune response rule (i.e. no repeated transpositions).

Here is Futurama’s proof of Sweet Clyde’s Inversion Theorem:

Recall that: Theorem.Every permutation of a finite set is a product of disjoint cycles. So if we can show the theorem for a k-cycle, then this is sufficient for the proof.

First, let p be a k-cycle: Introduce two new elements, x and y. We perform a sequence of transpositions. Consider (this is a special case of Sweet Clyde’s proof, and we use a slightly different notation).

We now see that whatever permutation p did, permutation s undoes: or Notice that x and y are interchanged, but all other elements are fixed.

So for each cycle in the permutation, x and y are interchanged. If the permutation consists of an even number of cycles, then x and y will be fixed. If there are an odd number of cycles in the permutation, then at the end, apply the permutation (x,y).

Oh yeah… the gratuitous cartoon nudity…

References • Fraleigh, John B., A First Course in Abstract Algebra, Third Edition, Addison-Wesley Publishing (1982). • Internet: The images in this presentation were shamelessly “borrowed” from the internet. Wikipedia and Google’s image search were of particular use. 3. Details on the proof are online at: http://theinfosphere.org/The_Prisoner_of_Benda

Permutation Groups: