ABSTRACT GENERALIZING the FUTURAMA THEOREM the 2010
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ABSTRACT GENERALIZING THE FUTURAMA THEOREM The 2010 episode of Futurama titled The Prisoner of Benda centers around a machine that swaps the brains of any two people who use it. The problem is, once two people use the machine to swap brains with each other, they cannot swap back. The author of the episode, mathematician Ken Keeler, uses Abstract Algebra to take the problem containing a set of swapped brains, and translate it into permutations in the group Sn. Keeler's method of solving the problem involves writing the inverse of the permutation as a product of transpositions that were not already used. The theorem and proof contained in this episode is known as The Futurama Theorem, or Keeler's Theorem. In 2014, it was proven that Keeler's method was the optimal solution to the problem. In this work, we will present a new proof of Keeler's Theorem. We will also generalize the theorem to products of larger cycles, starting with 3-cycles, and building up to our main goal: a solution for p-cycles where p is a prime. After this solution, we will use the same general ideas to create a solution using products of 2j-cycles. Jennifer E. Elder May 2016 GENERALIZING THE FUTURAMA THEOREM by Jennifer E. Elder A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Mathematics in the College of Science and Mathematics California State University, Fresno May 2016 APPROVED For the Department of Mathematics: We, the undersigned, certify that the thesis of the following student meets the required standards of scholarship, format, and style of the university and the student's graduate degree program for the awarding of the master's degree. Jennifer E. Elder Thesis Author Oscar Vega (Chair) Mathematics Stefaan Delcroix Mathematics Katherine Kelm Mathematics For the University Graduate Committee: Dean, Division of Graduate Studies AUTHORIZATION FOR REPRODUCTION OF MASTER'S THESIS I grant permission for the reproduction of this thesis in part or in its entirety without further authorization from me, on the condition that the person or agency requesting reproduction absorbs the cost and provides proper acknowledgment of authorship. X Permission to reproduce this thesis in part or in its entirety must be obtained from me. Signature of thesis author: ACKNOWLEDGMENTS First, I would like to thank my adviser, Dr. Vega, for his assistance and advice through the last two years of research. I am truly grateful for the time (and chocolate) that he provided, reading through everything I wrote, and working with me for so many hours each week as I progressed through this work. I have genuinely enjoyed working with him. I would also like to thank the members of my thesis committee, Dr. Delcroix and Dr. Kelm, for their time and input to help me make this a better paper. This would not have been possible without all of you. I would also like to thank everyone in the Math Department who have helped me over the last few years. In particular, I am grateful for the support from Dr. Caprau, Dr. Forg´acs, Elaina Aceves, and Kelsey Friesen. Their assistance has been invaluable, from working with me on Sonia Kovalevsky Day, traveling to conferences, studying for classes, and attending my talks on this project over, and over again. Finally, I would like to thank my family for being there for me over my entire college career. You put up with all my insane mutterings, and were patient as I wrote like I was running out of time. Your support and interest in my project have meant everything to me. The last two years would have been so much harder without your encouragement. Thank you. TABLE OF CONTENTS Page INTRODUCTION . 1 The Symmetric Group . 6 Keeler's Proof . 11 REPROVING KEELER'S THEOREM . 15 Examples . 15 Individual Cases . 16 A New Proof of Keeler's Theorem . 24 GENERALIZING TO PRODUCTS OF 3-CYCLES . 26 ANOTHER METHOD . 32 Revisiting the 3-Cycles . 32 Products of 5-cycles . 36 PRODUCTS OF p-CYCLES . 42 PRODUCTS OF 2j-CYCLES . 54 4-cycles . 54 2j- cycles . 59 CONCLUSIONS . 69 REFERENCES . 71 INTRODUCTION In the 2010 episode of Futurama, The Prisoner of Benda, Professor Farnsworth and Amy build a machine that can swap the brains of any two people. The two use the machine to swap brains with each other, but then discover that once two people have swapped with each other, they cannot swap back. It may not be immediately apparent, but this problem is directly connected to Abstract Algebra. Each brain swap can be described by a function in the group Sn. In order to understand the problem in this context, we started our research by looking at the basic definitions and examples about Sn presented in an undergraduate Abstract Algebra course. We worked our way from the basic definitions, through reproving the important theorems, up to the problems available in the, now classic, Ph.D.-level book Abstract Algebra by Dummit and Foote [3]. Once we were familiar with Sn, we began to write a new proof for Keeler's Theorem, which we will see in the next section. We used the insight this process gave us in order to generalize our process from products of transpositions to products of general cycles, as seen in the last two sections of this work. Before we see any of our solutions, we will present an example where some famous mathematicians discover a brain swapping machine, and use it. First, Sonia Kovalevsky and Hypathia use the machine, yielding the following situation. 2 ! Kovalevsky Hypatia Unfortunately, once they have swapped, they discover that they cannot swap back. So they ask other mathematicians to help them. After many additional swaps, we have the following positions described by the arrows. ! ! ! Noether Chatelet Hypatia Agnesi This means that Emmy Noether's brain is now in Emilie Chatelet's head, Emilie Chatelet's brain is in Hypatia's head, and Hypatia's brain is in Maria Agnesi's head. ! ! Agnesi Somerville Noether Maria Agnesi's brain is in Mary Somerville's head, and Mary Somerville's brain is in Emmy Noether's head. ! ! ! Kovalevsky Lovelace Germain Kovalevsky 3 Sonia Kovalevsky's brain is now in Ada Lovelace's head, Ada Lovelace's brain is in Sophie Germain's head, and Sophie Germain's brain is in Sonia Kovalevsky's head. We want to be able to transfer everyone back into their own heads, but the possible swaps are limited. Keeler's method of solving this problem involves bringing in two people who were not originally involved in the problem. Euler F ermat We start with the three person set of swaps, splitting the set in two pieces, with Germain and Lovelace in one set, and Kovalevsky in the other. And we start performing new swaps. ! F ermat Kovalevsky Now Fermat has Germain's Brain, and Kovalevsky has Fermat's brain. ! Euler Germain 4 After this swap, Euler has Lovelace's brain, and Germain has Euler's. ! F ermat Germain Now Germain has her own brain back, and Fermat has Euler's brain. ! Euler Lovelace After this swap, Lovelace has her own brain, and Euler has Kovalevsky's brain. ! Euler Kovalevsky Finally, Kovalevsky has her own brain, and Euler has Fermat's. If this was the end of the problem, we could swap Euler and Fermat back. However, we will work with the five person set of swaps, and switch the other two at the end if necessary. Once again, we split this set in two, with Noether, Chatelet and Hypatia in one set, and Agnesi and Somerville in the other. ! F ermat Agnesi 5 Now Fermat has Hypatia's brain, and Agnesi has Euler's brain. ! Euler Noether And now Euler has Somerville's brain, and Noether has Fermat's. ! F ermat Hypatia Hypatia has her own mind back, and Fermat has Chatelet's. ! F ermat Chatelet Chatelet has her own mind and Fermat now has Noether's brain. ! F ermat Noether Now Noether has her own mind, and Fermat has his own mind. We can start swapping with the second set of people now. 6 ! Euler Somerville Somerville has her brain back, and Euler has Agnesi's. ! Euler Agnesi And finally, with this last swap both Agnesi and Euler have their brains back. This solves the problem, since each participant has their own brain back. The Symmetric Group Now that we have seen an example, and how to solve the specific problem we had at hand, we need to introduce some definitions and notations that will be used in the rest of the work. Definition 1. Let A be a finite set containing n elements. We define SA as the set of bijections from A to itself. For simplicity's sake, we write this as A = f1; 2; : : : ; ng, and use the notation Sn. This is a group under composition of functions, and the elements in this set are called permutations. The identity for the group is denoted e. Definition 2. Let σ be a permutation in Sn. The order of σ is the smallest positive integer n such that σn = e. This integer is also denoted jσj. 7 There are different ways to write these functions, but we will use the most common method called cycle notation. Example 3. Suppose we define the following function in S4. 1 7! 2 2 7! 3 3 7! 1 4 7! 4: This function can be written as the following cycle: (1 2 3). Note that we do not write 4 as a part of this notation, since the function simply sends 4 back to itself.