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These notes are a wonderful introduction to group theory! Though only 39 pages in length, it covers a fair amount of the subject in a very clear fashion by relating the theory to the study of the Rubik’s Cube.
A Rubik’s Cube is a cube in which the 26 subcubes on the outside are internally hinged in such a way that rotation (by a quarter turn in either direction or a half turn) is possible in any plane of cubes. Each of the six sides is painted a distinct color, and the goal of the puzzle is to return the cube to a state in which each side has a single color after it has been randomized by repeated rotations. The puzzle was invented in the 1970s by the Hungarian Ernő Rubik and sold millions of copies worldwide over the next decade.
The number of possible positions of the Rubik’s cube is
(Turner and Gold 1985, Schönert). Hoey showed using the Pólya-Burnside lemma that there are positions up to conjugacy by whole-cube symmetries.
The group of operations on the Rubik’s cube is known as Rubik’s group, and the Cayley graph of that group is called Rubik’s graph. The minimum number of turns required to solve the cube from an arbitrary starting position is equal to the graph diameter of Rubik’s graph, and is sometimes known as God’s number. While algorithms exist for solving a cube from an arbitrary initial position, they are not necessarily optimal (i.e., requiring a minimum number of turns) and computation of God’s number is very difficult. It had been known since 1995 that a lower bound on the number of moves for the solution (in the worst case) was 20, it was not known until demonstrated by Rokicki et al. (2010) that no configuration requires more than 20 moves, thus establishing that God’s number is 20.
A Note to the Reader
These notes are based on a 2-week course that I taught for high school students at the Texas State Honors Summer Math Camp. All of the students in my class had taken elementary number theory at the camp, so I have assumed in these notes that readers are familiar with the integers mod n as well as the units mod n.
Because one goal of this class was a complete understanding of the Rubik’s cube, I have tried to use notation that makes discussing the Rubik’s cube as easy as possible. For example, I have chosen to use right group actions rather than left group actions.
The goal of these notes is to give an introduction to the subject of group theory, which is a branch of the mathematical area called algebra (or sometimes abstract algebra). You probably think of algebra as addition, multiplication, solving quadratic equations, and so on. Abstract algebra deals with all of this but, as the name suggests, in a much more abstract way! Rather than looking at a specific operation (like addition) on a specific set (like the set of real numbers, or the set of integers), abstract algebra is algebra done without really specifying what the operation or set is. This may be the first math you’ve encountered in which objects other than numbers are really studied!
A secondary goal of this class is to solve the Rubik’s cube. We will both develop methods for solving the
Rubik’s cube and prove (using group theory!) that our methods always enable us to solve the cube.
Douglas Hofstadter wrote an excellent introduction to the Rubik’s cube in the March 1981 issue of Scientific American. There are several books about the Rubik’s cube; my favorite is Inside Rubik’s Cube and Beyond by Christoph Bandelow. David Singmaster, who developed much of the usual notation for the Rubik’s cube, also has a book called Notes on Rubik’s ’Magic Cube,’ which I have not seen.
For an introduction to group theory, I recommend Abstract Algebra by I. N. Herstein. This is a wonderful book with wonderful exercises (and if you are new to group theory, you should do lots of the exercises). If you have some familiarity with group theory and want a good reference book, I recommend Abstract Algebra by David S. Dummit and Richard M. Foote.
Topics covered are:
- The Rubik’s Cube and Subgroups
- Cube Notation
- Making the Rubik’ Cube into a Group
- Simplifying Group Notation
- The Symmetric Group
- Disjoint Cycle Decomposition
- Rubik’s Cube
- Group Homomorphisms
- The Sign Homomorphisms
- The Alternating Group
- Group Actions
- Valid Configurations of the Rubik’s Cube
Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles, of Fermat’s Last Theorem. More concretely, an elliptic curve is the set of zeros of a cubic polynomial in two variables. Where is the equation of a general cubic polynomial. A famous example being
or in homogeneous form,
This is the first non-trivial case of Fermat’s Last Theorem.
A modular elliptic curve is an elliptic curve that admits a parametrization by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is a modular form in disguise.
In 1985, starting with a fictitious solution to Fermat’s last theorem (the Frey curve), G. Frey showed that he could create an unusual elliptic curve which appeared not to be modular. If the curve were not modular, then this would show that if Fermat’s last theorem were false, then the Taniyama-Shimura conjecture would also be false. Furthermore, if the Taniyama-Shimura conjecture is true, then so is Fermat’s last theorem.
However, Frey did not actually prove that his curve was not modular. The conjecture that Frey’s curve was not modular came to be called the “epsilon conjecture,” and was quickly proved by Ribet (Ribet’s theorem) in 1986, establishing a very close link between two mathematical structures (the Taniyama-Shimura conjecture and Fermat’s last theorem) which appeared previously to be completely unrelated
By proving the semistable case of the conjecture, Andrew Wiles proved Fermat’s Last Theorem.
Some Elliptic curves:
This is a 45 minute documentary [Fermat’s Last Theorem (1996)] about Andrew Wiles, who proved Fermat’s Last Theorem in 1994. The theorem was first conjectured by Pierre de Fermat in 1637 and baffled mathematicians for more than 350 years! It states that no three positive integers , , and can satisfy the equation for
The proof relied on elliptic curves and modular forms.
What do you think of this paper? This is work inspired from my friend Hei-Chi Chan. Read the proof here.