The Story of Z/6 +
Z/6 + did not know what it was. All of it’s friends had seemed to already know what they were and had already found their properties. Z/6 + had been told multiple times that a slow discovery was perfectly normal and that Z/6 + would eventually find itself. However 30 proofs had passed since that point and doubt had begun to creep up in everyone’s mind if now young adult Z/6 + would ever find what it was. Much speculation had been made throughout this time period in Z/6 +’s hometown, the land of polynomials. Some reasoned that since Z/6 +’s father, Z/6 was a set, and it’s mother, + was an operation, it made sense that Z/6 + would be a combination of the two. But what this combination was, nobody knew. So, Z/6 + decided that after 30 proofs with zero answers, that it would find the answer itself.
Z/6 + traveled a great many places across the mathematical world it lived in. It traveled across the great plains the fields. It traveled across the great ranges of the finite sets and smartly avoided the endless ranges of the infinite sets. Over its journey Z/6 + had discovered that the more it saw and learned, the more it’s questions grew. Why was it here in the mathematical world? What was it’s purpose? Z/6 + decided to ignore these questions and instead focusing on self discovery first. So Z/6 + traveled on, pushing through the various regions of its mathematical landscape, until finally reaching the land of Groups. Here Z/6 + met many great groups, including the symmetric groups and dihedral groups. It spent over 3 years with these groups, learning about them and their operations. It was on one fateful day, that Z/6 +’s great journey began. While visiting the group’s local planes, she met one of the elderly groups of the land, Z/ 6 * . Z/6 *, from the moment it met Z/6 +, knew that Z/6 + had to be a group. So Z/6 * took Z/6 + under it’s mentorship, determined to help Z/6 + find what it was, just as Z/6 *’s old mentor Z/5 * did.
In their first session, Z/6 + learned the ancient secrets of discovering groups. Z/6 * taught Z/6 + the following 4 ancient all important steps, passed down by generations of Groups :
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A group must have closure under its operation
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A group must have identity
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A group must have inversion
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A group must be associative under its operation
In their second session Z/6 * worked with Z/6 + to find its identity. After many hours of trial and deliberation, Z/6 + found that one element of itself that would let any element of Z/6 + be itself under the + operation. Z/6 + found 0.
After the identity was found, Z/6 *, realizing that the easiest part of this self exploration was complete, sent Z/6 + on its way, saying that the next three steps were to be done individually as tradition told.But Z/6 * gave Z/6 + a piece of paper, and told Z/6 + not to open it until completion of the journey, saying Z/6 + would understand then.
So Z/6 + set out in the wild world of groups, determined to find its inversion. Z/6 + traveled low and high across the mathematical world of groups.Tired, one day Z/6 + decided to rest under nice bodhi tree for a while. Z/6 + spent many days under the bodhi tree considering it’s inversion until finally realizing that its inversion was not in the external world, but rather was inside itself.. Examining itself closely. Z/6 + realized that for every element that it had, it also had that element's inverse. For the first time in Z/6 +’s life, it used its operation to test this theory. It had heard about the use of a group's operation, but it had never occurred to Z/6 + to use it. Sure enough, every element had an inverse. Every element she had, there was always another element that when combined with the operation, would always return back to its identity. Motivated, Z/6 + continued to the next step, closure. This step proved easy to discover, as no matter what element of itself that Z/6 + seemed to use with its operation, it always got an element that it had. Excitement started to build, as Z/6 + moved closer and closer to becoming a group. Z/6 + read the final requirement for a group and it’s heart sank. “The group must be associative under its operation” was the final requirement. But Z/6 + did not know what associative meant only where it could find what it meant. Z/6 + had heard rumors about this terrible and confusing land. But Z/6 + was determined to find what it was, to discover itself. Moving away from the bodhi tree, Z/6 + began its long and treacherous journey to the land of number theory.
It took three proofs for Z/6+ to reach the land of number theory. It was a brand new world for Z/6 +, seeing for the first time in it’s short mathematical life prime numbers and weird numbers that would call themselves fermat’s numbers. Although Z/6 + was new to the world, it pressed on to find what associative meant. Z/6 + arrived at an ancient library and entered, anxiously hoping that the answers to it’s questions would be inside the library. Finally Z/6 + stumbled upon a book that would have the answers to all of it’s questions. It opened the book and read the definition of associativity. It read:
a+(b+c) = (a+b)+c.
Using this definition Z/6 + performed the addition operation on itself using its own elements. After rechecking multiple time, Z/6 +, much to the dismay of the librarians, ran out of the library yelling in happiness. Z/6 + finally find out what it was, it was a group. But after the initial rush of happiness subsided, Z/6 + realized that it felt no more fulfilled than it did before it’s journey of self discovery. The questions of why it was here and what it’s purpose was were still left unquestioned. Instead of the joy of self exploration, Z/6 + felt a deep emptiness inside.
So Z/6 + traveled back to the land of the groups, hoping to find its mentor. But upon arrival, Z/6 + found out that Z/6 * had traveled to the outskirts of the land of groups to start a new city, a city for Abelian groups Z/6 +, remembering the note that Z/6 * left, opened it and inside was the last secret requirement that applied for some groups. It was that some groups could be commutative. This mean that the order of the elements did not matter for an operation. Anxiously Z/6 + tested this final requirement and found that it was in fact commutative. Z/6 + finally new what it was, it was an abelian group. The note instructruction Z/6 + to stay in the land of groups and help other groups discover themselves, as Z/6 * had done for Z/6 +.
So Z/6 + would stay in the land of groups and would send other groups on their journey of self discovery, just as it’s mentor had done. But the empty feeling inside it had since it’s visit to the land of number theory remained. Over time Z/6 + realized that the emptiness it had inside was due to the fact that Z/6 + knew that no matter how many other questions it found the answer to, it would never find the answers to the questions it really wanted answered. Z/6 + realized that the answers to the question of why it was here and what it’s purpose was, were simply questions that had no answers. This knowledge of the absurdity of attempting to answer these existential questions that had no answers kept Z/6 + from spending any significant time on these questions. But the emptiness that it had inside stayed within it for a very long time.