### Errata and Comments Page for 1st printing of Adventures in group theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys

Errata which have been fixed in the 2nd printing are marked with a *. To the best of my knowledge, all errata are fixed in the 2nd edition.
News: The 2nd edition is now available at amazon.com or from the Johns Hopkins University Press. (All royalties go directly to charity - half to the Sage Foundation to support open-source mathematical software development, and half to the Earth Island Institute, an environmental organization with projects all over the world.)

Thanks to those readers who emailed or snail-mailed me some of the following errata. In particular, I'd like to thank Jamie Adams, Lewis Nowitz, David Youd, Roger Johnson, Jaap Scherphuis, Michael Hoy, Tom Davis, John Rood, Trevor Irwin, Stephen Lepp, Mark Edwards, Carl Patterson, Peter Neumann (seven pages!), Bill Zeno, Herbert Kociemba, Alastair Farrugia, Matthew Lewis, and Christopher Tuffley.

• Page 6, line -7: Add: "A finite set is one which can be enumerated by a finite sequence.
• On page 13, line -8: two axioms of a vector space are missing. Insert in place of the period at the end of (d), ", (e) the associative law: u+(v+w)=(u+v)+w, for all u,v,w in V, (f) 1*v = v and a(b*v)=(ab)*v, for all v in V, a, b in R."
• On page 14, lines 1-3: Also must assume for all s in S there exists t in T with (s,t) in X.
• On page 22: Definition 2.3.3 (on page 23) should go before Example 2.3.4.
• On page 24, line 5, the interval [0,1) is undefined. In general,
[a,b) = {x real | a <= x < b}.
• * The example on page 26, lines 16-20, is correct but illustrates the "addition principle" not the "multiplication principle", as it should. To get example of the "multiplication principle", change "Exactly one" on line 19 to "Each one" and "30" on line 20 to "103=1000". Also, all insults are assumed to be different.
• On page 27, line 8: The condition m < n is unnecessary and should be deleted.
• In Example 2.4.5 on page 27, the use of "poker hand" is misleading, since (as the example clearly states) we are counting ordered 5-tuples of 5 cards from an ordinary card deck. In the comman usage "poker hands" are considered as being unordered since their value in the game does not depend on their arrangement in your hand. I hope this causes no confusion in illustrating the formula for counting permutations. There is also a typo in the formula given, which is corrected below.
• On page 33, Ponderable 3.1.4: Here the composition is as functions.
• On page 36, line 4+: Here the orientation of the "axes" of the matrix coordinates is: 1st coordinate goes down, 2nd coordinate goes across. This is not to be confused with the orientation of the "axes" in the xy-plane.
• On page 36, line 9: Here the symmetric group Sn has not been defined. See page 70.
• On page 47-48. Ponderable 4.1.1 and Ponderable 4.1.2 are identical. Delete Ponderable 4.1.1.
• On page 48, line 9, T is undefined. It is defined as the set of puzzle pieces.
• * On page 48, line 3, Z/nZ is undefined. It is defined on the bottom of page 67.
• page 50: The diagram is flipped (should be numbered clockwise).
• On page 51, theta and phi are undefined. They are the usual angles in spherical coordinates.
• On pages 55, 56, 63, 116, 119, 177. Not an errata but unexplained shorthand notation: instead of writing the "2x2x2 cube puzzle" (or "3x3x3 ..."), I simply say "2x2 cube puzzle" (or "3x3 ...").
• * page 67: The diagram on this page is wrong. The arrows should point in a clock-wise manner, so the bottom-most arrow should be turned around. The right-most vertex should be a "j" (not "k") and the left-most vertex should be a "k" (not "j").
• Page 77, line -3: Insert "Another version of the following result will be given in Theorem 5.11.1 below."
• pages 77 and 95. It is worth remarking that, contrary to what might be suggested by the name of the result, Lagrange's theorem, in the forms stated here, is not due to Lagrange. In the form given in the book, Lagrange's theorem (for permutation groups) was probably known to Galois and appeared in a work of Serret and of Jordan. See also Oxford's project page for students
• Page 95, lines 8-10: The last part of the sketch of the proof of Corollary 5.11.2 is wrong. Corrections below. At the end of the proof (brefore the square denoting the end of the proof), add "For details, see for example, Theorem 10.2 in [G]." In fact, Theorem 10.2 is on the internet, though there it is called Theorem 10.1.2.
• Page 95, 13: Though not a typo, it is inaccurate to refer to this result as Lagrange's. Better is to replace "(Lagrange)" by "(Lagrange's Theorem)".
• Page 95, lines 20-21: Though not a typo, this result has been given enough times that it shouldn't be labeled as a corollary. So, replace, "Corollary 5.11.3: If ... of G." by As a consequence of Lagranges' therem, if ... of G."
• page 100, section 6.2.1. This section uses graph-theoretic terminology which is not introduced until section 7.1.
• * page 101. Though very similar, the original Parker Brothers Merlin Magic game did not have exactly the same moves as a 3x3 Lights Out. Pressing a corner changes the centre light whereas pressing an edge does not. It does not have a symmetric toggle matrix, and its graph is a directed graph. For other discussion, see Jaap's pages and Matthew Baker's site. In the second printing, this oversight is corrected.
• * page 112, figure 6.5.5: The outer pentagon (numbers 7-11) should be rotated clockwise 36 degrees with respect to the rest.
• * page 114: The keychain light's out "toggle matrix" T is wrong: The correct one is:
[1  1  0  1  1  0  0  0  0  0  0  0  1  0  0  0]
[1  1  1  0  0  1  0  0  0  0  0  0  0  1  0  0]
[0  1  1  1  0  0  1  0  0  0  0  0  0  0  1  0]
[1  0  1  1  0  0  0  1  0  0  0  0  0  0  0  1]
[1  0  0  0  1  1  0  1  1  0  0  0  0  0  0  0]
[0  1  0  0  1  1  1  0  0  1  0  0  0  0  0  0]
[0  0  1  0  0  1  1  1  0  0  1  0  0  0  0  0]
[0  0  0  1  1  0  1  1  0  0  0  1  0  0  0  0]
[0  0  0  0  1  0  0  0  1  1  0  1  1  0  0  0]
[0  0  0  0  0  1  0  0  1  1  1  0  0  1  0  0]
[0  0  0  0  0  0  1  0  0  1  1  1  0  0  1  0]
[0  0  0  0  0  0  0  1  1  0  1  1  0  0  0  1]
[1  0  0  0  0  0  0  0  1  0  0  0  1  1  0  1]
[0  1  0  0  0  0  0  0  0  1  0  0  1  1  1  0]
[0  0  1  0  0  0  0  0  0  0  1  0  0  1  1  1]
[0  0  0  1  0  0  0  0  0  0  0  1  1  0  1  1]

The rank of this matrix T over the field Z/2Z is 16 (not 15, as stated on page 114, line -5), so it is invertible. The Lemma 6.5.4 should read: The codimension is 0. The keychain lighht's out puzzle is always solvable.
• Page 115: Not a typo, but there is no need to assume that the sets of vertices and edges are countable,
• Page 117, lines 2-3: Not a typo but there is no need to assume that G is a permutation group to define a Cayley graph.
• Page 122. The paragraph "we can consider..." should be rewritten. The definition of simple path can be omitted. (It's not used.) The following definition should be added. If p,p' are closed paths based at x0 then the composition pp'is the path obtained by juxtaposing the path for p with that for p'.
• Page 127, lines 2-4: Instead of calling Jordan a founder of group theory, calling him a pioneer might be more accurate.
• Page 129, line3 3-4: Not wrong, but the statement "In particular, ... invertible." is too complicated. Better would be "The matrix A is invertible since, by definition, A-1 = tA."
• Page 138, Ponderable 9.1.4: The direct product Cm x Cn is not defined until Definition 9.6.1.
• Page139, Theorem 9.2.1: A permutation group G acting on a finite set X means that G is a subgroup of SX.
• Page 140, line -11: Not a typo but a comment: The precise reference to [NST] is [NST], Table 15.7. When we say "regards as 3-dimensional transformations", we mean, as elements of O(3,R).
• Page 144, line -4: Not a typo but a comment: Another example: any subgroup of S6 of order 2 is non-normal.
• * On the bottom of page 156 the symbol for the semi-direct product is missing a subscript from the notation. These math symbols are not available in html, but the entry in the table below gives a good indication of the the correction is.
• * The dicyclic or generalized quaternion group Qn, used in the tables on page 169-172, is undefined. It is
Q2m= < a,b | an=1, aba=b, b2=am >,
for any integer m>1.
• * Some of the semi-direct products indicated in the tables on pages 169-172 are wrong. Generally, if the table says "semi-direct product of A with B (with B normal)", it should read "semi-direct product of A with B (with A normal)". This occurs on
page 170, lines -5, -8, -11
page 171, lines 7, 21, 26
page 172, lines 18, 21, 24
• Page 172, line -14: Before Ponderable 10.4.1, add "Let G denote the Rubik's cube group."
• This is not an errata but an additional comment: On page 179, the labeling of the corners used in Example 11.1.1 are as in Theorem 9.6.1. They are given by the following diagrams:
• This is not an errata but an additional comment: On page 180, the labeling of the edges used in Example 11.1.2 are as in Theorem 9.6.1. They are given by the following diagrams:
• * On page 180, in the top paragraph section 11.1.3, change every occurrence of the word "corner" with edge (5 times), replace "vertices" with "edges" (1 time), and "vertex" with "edge" (1 time).
• * page 180. The table in Example 11.1.2 has a typo. The entry for "R" is wrong. It should be:
(0,1,0,0,0,0,0,0,0,1,0,0)
• pages 191-192. Eqn (11.5): some of the powers of 2 are wrong and there is a missing term of "2^11 -1". The total is 15687871. Eqns (11.6) and (11.7): this is a subset of the terms in (11.5) and the same changes must be made to the powers of 2. In (11.6) the correct total is 8080447. In (11.7) the correct total is 7607424. The total number of elements of order 2 is 170911549183. I thank Herbert Kociemba for help in fixing this.
• page 195. Here I claimed David Singmaster proved < F,U > isom S7 x PGL2(F5). Not only did he not prove this, it's false. I meant to say that the action of the group < F,U > on the edges is isomorphic to S7 and the action of < F,U > on the corners is isomorphic to PGL2(F5).

My apologies to David Singmaster.

• page 217: For those unfamiliar with it, the square one puzzle is described on Jaap's web page (back to) square 1.
• page 217, lines 14-16: Theorem 13.5.1 should be restated as follows: Theorem 13.5.1: Let G1 be as above and let G be the subgroup generated by the "square moves" d3, u3, B(u3,1), B(1,d3), T(u3,1), and T(1,d3), as below. The G1 is isomorphic to S8 x S8. G is isomorphic to to the kernel of index 2 in G1 = S8 x S8 of the homomorphism f : S8 x S8 --> {+1,-1} defined by f(g,h) = sgn(g)sgn(h). Consequently, |G1| = 214345272 and |G| = 213345272.
• * On page 244, the solution to the 5x5 lights out is incomplete. The row 1 states s15, s134, s235, s1245 (which are missing from the book) are also solvable.

They can be solved by using x15=

 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0
x134=  0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 0 0
x235=  0 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0 0
x1245=  0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 0

• pages 252, 253: There are two references which should be added: Jaap's puzzle page and the excellent book by Dixon and Mortimer, Permutation groups.
[Ja] Jaap Scherphuis, Jaap's puzzle page, available on the www at the url www.geocities.com/jaapsch/puzzles/.
[DiMo] J. Dixon and B. Mortimer, Permutation groups, Springer-Verlag, 1996.
 page, line read should be ix, 8 the previous a previous xiii, -8 solved was motivated by 6, -6 not a subset of a finite set not finite 6, -1 it's its 14, 11 fg fog = fg 14, -6 |S| = |T|. |S| = |T| < infinity.. 15, 8 f:S --> Z f:Z-->S 16, -9,-8 linear transformations matrices 19, -11 k = m k = m+1 19, -11 a_{ij} (AB)_{ij} 19, -9 row of column of 20, -1 det(Aij) det(Aij)aij 21, 2 or and 22, -3 21 23 22, -18 (s,t) belongs to S (s,t) belongs to R *26, 19 Exactly one Each one *26, 20 30 103=1000 *26, -9 and an *26, -3 an and 27, 12 object objects 27, -12 n, n-1, ..., n-m n, n-1, ..., n-m+1 27, -10 poker hands "ordered poker hands" 27, -8 52*51*50*49*48*47 = 14658134400 = (1.4...)x1010 52*51*50*49*48 = 311875200 = (3.1...)x108 28, 5 an combination a combination 28, 13-14 right ... left left ... right 31, 15 Ss Sc 33, -13 a point of points a collection of points 33, -6 It's Its 37, 10 chapter 8 chapter 9 (see Theorem 9.3.1) 37, 15 then are distinct then 38, 15 Si Ai 43, -18 (n,an) (n,an-1) 42, 9-10 row column 43, -14 (bn,n) (bn-1,n) *45, 23 a(ab)2 (ab)2a *45, 26 a(ab)2 (ab)2a *46, 5 ka(ab)2 k(ab)2a *46, 7 k2a(ab)2 k2(ab)2a 47, -1 five four 50, -10 radians (degrees) 51, 2 closed geodesic path closed path *52, -9 f1 f1r1 *61, 18 120 degrees 72 degrees 67, 1 {i,j,k}. {i,j,k}, and x doesn't equal y. 68, -9 D2n Dn 70, -5 o : G x G x G G x G ---> G 72, -9 What didn't Why didn't 76, 16 such that and a place token p_{i+1} such that *76, -3 {1,3,6} {1,3,4} *76, -2 {3,6} {3,4} 77, -2 (Lagrange) ("Lagrange's Theorem") 78, 7 x in G. x in G. (Exercise: verify this.) 78, 8 "so" "(see Ponderable 2.3.3), so" *78, -9 "...belongs to G" "...belongs to Z(G)" 79, 13 identifies identified 79, -4 likelyhood likelihood *80, 4 MD2 MF2 *80, -6 ...showed to there ... ...showed that there ... *82, top diagram label "8" (resp., "6") on cube label "6" (resp., "8") on cube 84, -14 ... substitutions ... des substitutions 84, -3 gh=1 gh=g 86, 5 g in G_* d \geq 0 *87, 2 clases classes *87, 10-11 g*x*g^{-1} g*y*g^{-1} 88, -10 left action right action 88, -5 right left 88, -3 on the right on the left 89, 5 right action left action 89, 15 send sent 89, 18 are induce 89, 19 (though not all are)! . 89, 21 In other words, Roughly speaking, 89, 21-22 group acting on a set at random' random` finite group acting primitively on a finite set 89, 24 corners corner 89, -1, -2 (a), (b) (b), (c) 91, 18 set of consisting set consisting *92, -6 right face down face 93, 17 sbgroup subgroup 93, lines 24,25 groups subgroups 94, 15 both and both an 95, 8-9 Then ... G = H* ... maximal). Thus p either ... Then G = H* ... maximal). This implies p either ... 95, 10 In either case, the result ... hypothesis. The result ... hypothesis. 95, 13 (Lagrange) ("Lagrange's Theorem") *96, 13 a(ab)2 (ab)2a 99, 8 organisational organizational 99, -13 choosen chosen 99, -11 always be ... is), be ... is) is 1, *99, -7 .5 1 99, -7 always be 99, -6 always be solved. be solved is 1. 100, 18 and shall and *101, 14 (M-1)(N-1) N(M-1)+M(N-1) *101, section 6.3.1 Merlin's magic 3x3 light's out 104, 5 for some provided *104, 7 Ei,j in MN Ei,j in MMxN *105, -8 omit "Merlin's magic and" 105, 3 2 3 105, 4 3 4 105, 5 4 5 106, -7 = 0 \not= 0 107, -20 effectivly effectively 107, -14 intorduce introduce 107, -9 \cdot F \cdot : F 107, -4,-3 the vector \vec{0}=(0,0,...,0)\in V satisfies there is a vector \vec{0}\in V satisfying 109, 15 if multiplicity of multiplicity 109, -16 matrix square diagonalizable matrix 110, 17 is upper-triangular. The number of non-zero entries on the diagonal is reduced. The number of non-zero rows 111, -2 reculangular rectangular 111, -1 Aien Alien *112, figure 6.5.5 The outer pentagon (numbers 7-11) should be rotated clockwise 36 degrees with respect to the rest. *114, 10 matrix T see above *114, -5 15 16 *114, -3 is 1 is 0 *114, -2 are initial are no initial 116, 11 eminating emanating 117, 8 who whom 117, 9 laywer lawyer 117, -6 v's v *117, -2 (1 2) , (2 3) (1,2) , (2,3) 119, -21 Micheal Michael 120, -12 who whom 122, 6 We can ... path p, Let a path p on puz(Gamma) to be a sequence of moves 122, 7 n <= 1 n >= 1 122, 14 of paths based of closed paths based *122, 12 14-15 puzzle 15 puzzle 123, 1 has has at most 125, 9 vertices edges 126, line -7,-6,-5 symmetric symmetry 127, -12 (1/2)(...||^2 + ||.. (1/4)(...||^2 - ||.. 127, -1 transpose identity. transpose 128, 9 *B *A 128, 10 if an if and 129, 3-4 In particular, ... invertible. The matrix A is invertible since, by definition, its inverse is ^tA. 132, 8 permutations of permutations in 132, 17 permutations of permutations in 132, 17 A5 x S5 A5 x C2 138, 1 E. Jordan C. Jordan 139, 12 X x X X x X - Delta (where Delta = {(x,x) | x in X}) 139, 17 belonging to X belonging to X, xi <> xj 139, 17 for each provided 139, -15 group permutation group 139, -11 [R] [DiMo], Theorem 7.6A 141, lines 4,5 , and in ... i,j,k. , as in section 5.1. *141, 20,21 left, right right, left (resp.) *141, 23 Let g Let x 141, -8 concept concepts 141, lines -7, -6 finite groups finite simple groups 142, 9 three cubes two cubes 142, -14 for all 1 <= i <= n provided 1 <= i <= n 142, -1 for all 1 <= j <= n provided 1 <= j <= n 143, 2 <= 1 <= n 145, 4 E. Galois Ruffini and Abel 146, 11 who whom 146, -15 normal groups normal subgroups 147, -21 isomomorphism isomorphism 147, -1 isomorphism between isomorphism 150, -14 H acts on X. Note Note, H acts on X and 150, -4 is an element are elements 150, -3 any element other element of E each other *156, 3 somwhat somewhat *156, -1 H1 x H2 H1 xphi H2 157, -19 colun column 158, -2,-1 (x,y) g 158, -1 (-x,-y) -g 161, 10 group arose group, originally arose *163, -14 n n-1 *163, -12 n=2 n=3 164, 15 Rubik's cube. Rubik's cube which does not contain a basic move followed by its inverse 165, 9 , however . However, 165, 18 occurance occurence, 165, -10 cardiality cardinality 166, -10 who whom 166, -6 R The letter script-R 166, -5 realtions relations 166, -3 relations R relations script-R *170, 4-5 should be a horizonal line separating cases 15,16 172, -13 (a) . 172, -11,-10 , and in ... {i,j,k}. , as in section 5.1. *178, 1 lu, and bu lu, bl, and bu *180, 3, 4, 7, 8 corner edge *180, 4 vertex edge *180, 5 vertices edges *180, 17 (0,1,0,0,0,1,1,0,0,1,0,0) (0,1,0,0,0,0,0,0,0,1,0,0) 180, -1 (SV wr C38)x(SE wr C212) (C3 wrSV )x(C2 wr SE) 181, 1-2 C38xSVx C38xSV C38xSVx C212xSE 185, 12-13 allows us to count its elements easier. enables us to determine the size of this group. 186, -1 isomomorphism isomorphism 188, -2 of (non-trivial) of 189, 9 haiku poem 193, 8 x1e1 +...xn en x1e1 +...+xn en 194, 15 More generally, let Let 194, lines 21, -10 e2i-2 e2i-1 195, 6 < F,U > .... < sigma(F),sigma(U) > isom S7, < rho(F),rho(U) > isom PGL2(F5), 198, -6 fractions fractional 201, 3 enties entries 201, lines -12, -14, -16 polyhedra polyhedron 205, 7 undeground underground 215, -13 SE wr C2 C2 wr SE 216, -2 G G_1 216, -2 G G_1 217, 1 and 4 G G_1 217, 14-16 Theorem 13.5.1 see above 217, 18 possible possible in G 217, -9 up down 223, -15 isocahedron icosahedron 224, -13 in his PhD thesis. a year after his Docteur es Sciences thesis in 1859. 225, -18 stating containing 225, -11 A8, A12 A7, A11 229, -8 Matheiu Mathieu 230, lines -18,-17,-7 isocahedron icosahedron 230, -14 many experts thought they had classified all finite simple groups some experts thought all finite simple groups had been classified 231, lines 7,8,11 parallelpiped parallelepiped 231, 20 giving the first proof giving one of the first proofs 232, -1 [R], ch. 9 [DM], Theorem 7.6A 251, -11 Berlekamp E. Berlekamp *253, 17 D. Hofstatler D. Hofstater *253, 17 Metamathematical Themas Metamagical Themas 259, 14 E. Jordan C. Jordan

Last Updated on 2009-6-8

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