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. Becauseone 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.
Here is some notation that will be used throughout.
Z N Q R Z/nZ (Z/nZ)
the the the the the the
set set set set setset
of of of of of of
integers . . . , −3, −2, −1, 0, 1, 2, 3, . . . positive integers 1, 2, 3, . . . rational numbers (fractions) real numbers integers mod n units mod n
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 mathyou’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 ScientiﬁcAmerican. 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 withwonderful 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.
To understand the Rubik’s cube properly, we ﬁrst need to talk about some diﬀerent properties of functions. Deﬁnition 1.1. A functionor map f from a domain D to a range R (we write f : D → R) is a rule which assigns to each element x ∈ D a unique element y ∈ R. We write f (x) = y. We say that y is the image of x and that x is a preimage of y. Note that an element in D has exactly one image, but an element of R may have 0, 1, or more than 1 preimage. Example 1.2. We can deﬁne a function f : → by f (x) = x2 . If x is any realnumber, its image is the √ √ real number x2 . On the other hand, if y is a positive real number, it has two preimages, y and − y. The real number 0 has a single preimage, 0; negative numbers have no preimages. y Functions will provide important examples of groups later on; we will also use functions to “translate” information from one group to another. Deﬁnition 1.3. A function f : D → R is calledone-to-one if x1 = x2 implies f (x1 ) = f (x2 ) for x1 , x2 ∈ D. That is, each element of R has at most one preimage. Example 1.4. Consider the function f : → deﬁned by f (x) = x + 1. This function is one-to-one since, if x1 = x2 , then x1 + 1 = x2 + 1. If x ∈ is an integer, then it has a single preimage (namely, x − 1). If x ∈ is not an integer, then it has no preimage.
Z R R...