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Adventures in Group Theory

Johns Hopkins University Press
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Adventures in Group Theory is a tour through the algebra of several "permutation puzzles." Although the main focus is on the Rubik's Cube, several other puzzles are explored to a lesser degree. Joyner pulls together a great deal of information about the mathematics of the Rubik's Cube from various sources and mixes it with a lot of algebra. (And I mean a lot!) The book has the following layout:

Chapter 1 An introduction to Boolean algebra, logic, and the basic moves of the Rubik's cube (using the Singmaster notation: U D L R F B).

Chapter 2 Functions, Cartesian products, vector spaces, matrices, determinants, relations and equivalence classes, and some basic combinatorics.

Chapter 3 Permutations, signs of permutations, permutation matrices (a little on permutations of the edges of the Rubik's cube), and the symmetry groups Sn.

Chapter 4 Permutation puzzles are introduced: Sam Loyd's 15 puzzle, the Hockeypuck puzzle, the Rainbow Masterball, the Pyraminx, the 2x2 and 3x3 Rubik's Cube, the Skewb, the Megaminx, and several others.

Chapter 5 An introduction to group theory. The quaternions, finite cyclic groups, dihedral groups, and symmetric groups are presented. The superflip is mentioned (this element generates the center of the Rubik's Cube Group). Basic properties of groups, commutators, the notion of conjugacy, orbits and group actions, and cosets

Chapter 6 A study of Merlin's Machine and some variants using matrices. Some linear algebra is presented.

Chapter 7 Graphs, with special attention to the Cayley graph associated with a group. The problem of God's Algorithm is presented: given a Cayley graph associated to a group, find an effective, practical algorithm for finding a path from a given vertex to the vertex corresponding to the identity. In practical terms, this would be an algorithm which could take any position of the Rubik's Cube and find the "quickest" solution.

Chapter 8 Platonic Solids and the notion of the symmetries of three-dimensional space.

Chapter 9 Group homomorphisms, isomorphisms, automorphisms, and actions are introduced as a way of determining when two groups are in fact the same (i.e. isomorphic). We also get quotient groups, and the three fundamental isomorphism theorems of groups. The direct product of groups is given as a way of examining the effects of permutations on the Rubik's cube. (Cubes go to cubes and edges go to edges.) We get the First Fundamental Theorem of Rubik's Cube Theory. The remainder of the chapter deals with semi-direct products and wreath products.

Chapter 10 Free Groups are introduced, along with the concept of words. Cayley graphs are mentioned again, with Poincaré polynomials. A description of all groups up to order 25 is given.

Chapter 11 Giving the cube an orientation, we find the group of all possible moves of the Rubik's Cube group (the legal Rubik's Cube group). The group structure is given (the direct product of two groups: the semi-direct product of the permutation group SVof the corners with the eighth power of the cyclic group of order three [this determines the possible permutations of the eight corners] and the semi-direct product of the permutation group SE of the edges with the twelfth power of the cyclic group of order two [this determine the possible permutations of the twelve edges], modulo three conditions on the edges and vertices). The number of elements of order two in the Rubik's Cube group is counted using a combinatorial argument (there are 334,864,275,867).

Chapter 12 Various subgroups of the Rubik's Cube group are studied. Some of this is achieved with the help of finite fields. Projective General Linear Groups are used.

Chapter 13 General results about other permutation puzzles.

Chapter 14 Some connections between the Mathieu group, coding theory, and Hadamard matrices are explored.

Chapter 15 Several solution strategies for the Rubik's Cube are presented in very general terms, as well as some strategies for other permutation groups.

Chapter 16 A summary of questions that are currently unanswered.

If you like puzzles, this is a somewhat fun book. If you like algebra, this is a fun book. If you like puzzles and algebra (which is the category that I fall into), this is a really fun book. Joyner introduces a great deal of algebra in this book. Most of it is undergraduate level, although some borders on graduate level material. If you've had a couple of semesters of algebra, you'll probably find the book entertaining, mathematically speaking. If you haven't, then you won't get that much out. While the algebra is (mostly) self-contained, it is presented quickly, and the author gives just enough to do what he wants. So don't expect it to read like an algebra textbook.

One of the most fun aspects of the book is the interesting (from the algebraic perspective) collection of trivia concerning the Rubik's Cube Group. For example, the structure of this group, the center of the Rubik's Cube Group, the structure of some of the subgroups of the Rubik's Cube Group, including an embedding of the quaternions into the group, and an example of two elements which generate the whole Rubik's Cube group. Joyner does a good job explaining how the structure of the Rubik's Cube works. But again, the algebraic description requires some semi-advanced algebra. The audience seems to be mathematicians with a fairly good knowledge of algebra and an interest in permutation puzzles. If you fall into that category, this book is for you.

Donald L. Vestal is an Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at

Date Received: 
Thursday, June 6, 2002
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David Joyner
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Donald L. Vestal
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Thursday, October 11, 2007