Errata for Adventures in Group Theory

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.

General comments:

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 S_n 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 C_m x C_n 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 S_X.
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 Q_n, used in the tables on page 169-172, is undefined. It is Q_2m= < a,b | aⁿ=1, aba=b, b²=a^m >, 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(A_ij)	det(A_ij)a_ij
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	5251504948*47 = 14658134400 = (1.4...)x10¹⁰	5251504948 = 311875200 = (3.1...)x10⁸
28, 5	an combination	a combination
28, 13-14	right ... left	left ... right
31, 15	S^s	S^c
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	S_i	A_i
43, -18	(n,a_n)	(n,a_n-1)
42, 9-10	row	column
43, -14	(b_n,n)	(b_n-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	D_2n	D_n
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	M_D²	M_F²
*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	g^h=1	g^h=g
86, 5	g in G_*	d \geq 0
*87, 2	clases	classes
*87, 10-11	gxg^{-1}	gyg^{-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	E_i,j in M_N	E_i,j in M_MxN
*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, x_i <> x_j
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 x_phi 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	(S_V wr C₃⁸)x(S_E wr C₂¹²)	(C₃ wrS_V )x(C₂ wr S_E)
181, 1-2	C₃⁸xS_Vx C₃⁸xS_V	C₃⁸xS_Vx C₂¹²xS_E
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 +...x_n e_n	x₁e₁ +...+x_n e_n
194, 15	More generally, let	Let
194, lines 21, -10	e₂^i-2	e₂^i-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	S_E wr C₂	C₂ wr S_E
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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