Errata
If you think you've found an error in the textbook, send me an email. If you're correct, I'll post a description here as soon as possible, to eliminate potential confusion for other readers. Special thanks to Alan Reynolds, for sending me helpful and clear reports of errata in the book, accounting for much of the content of this page.
Chapter 3
Page 26, after Definition 3.1
The fourth line of the second paragraph after Definition 3.1 says "Note that new each state..." when it should say "Note that each new state..."
Page 29, Figure 3.5
The figure has two identical shapes in it, when the one on the right should actually be flipped about a vertical axis. A corrected version of the figure is shown below.
Page 40, Exercise 3.12
Part (c) does not have the same symmetries as any of the figures in the previous exercise; it has the same symmetries as the frieze pattern from Section 3.1.3.
Chapter 4
Page 52, Exercise 4.3
This exercise mentions "commutative" without defining it. It is defined a few pages later, in Exercise 4.23.
Page 54, Exercise 4.6(b), line 1
It should read "on the right of Exercise 4.4," rather than Exercise 4.5.
Page 60, Exercise 4.26(c) and Page 61, Exercises 4.29 and 4.30
The element naming convention does not match that of the Cayley diagram for A4 on page 54. The elements a1, a2, b1, and b2 should be a, a2, b, and b2 instead, respectively. It is probably easier to rename the elements in the figure than in the exercises, but either way will fix the problem. Here is an example for how you could correct Exercise 4.26.
Chapter 5
Page 69, second paragraph
The paragraph mentions Exercise 4.6 as stating that Cayley diagram arrows represent multiplication on the right. Although arrows do indeed mean right multiplication, Exercise 4.6 does not actually say so. The first place such a statement seems to appear is in Section 4.2, but it is not explicit. Exercise 4.6 should probably be enhanced to say what this paragraph claims that it says.
Page 71, last lines
The phrase "the right of Figure 5.9" should say "the left of Figure 5.9."
Page 94, Exercise 5.38
The term "order" is used in this exercise but not defined. Thus this exercise belongs later in the book, after that term has been defined. (You can find its definition on page 110, in Exercise 6.11.)
Chapter 7
Page 133, Figure 7.20
The elements in the bottom row of each copy of C6 are labeled incorrectly. Since the blue generator indicates the action of adding 3 mod 6, the figure should instead look like the one shown here.
Page 140, third line of second-to-last paragraph
The notation NV4(A4) should instead be NA4(V4).
Page 155, Exercise 7.37
It should say taht Figure 7.33 shows that a and b are conjugates, not a and c.
Chapter 8
Page 163, Figure 8.8
Two elements in the group of complex numbers are missing their negative signs. The figure should instead look like the one shown here.
Page 184, Exercise 8.24
The equation should read an = bm + 1.
Pages 185-186, Exercises 8.29 and 8.30
Throughout these exercises, the semidirect products are written in the wrong order. For instance, in 8.29 part (a), because θ : C2 → Aut(C5), it should ask about the smidirect product of C5 with C2, not the other way around. The same error is repeated in parts (c) and (d); in each case, the homomorphisms are correct and the product notation is backwards. In 8.30 it should speak of the product of C3 with C4.
Page 187, Exercise 8.40
It should ask you to find an isomorphism between Q+×C2 and Q*, not between Q×C2 and Q*.
Page 199, first displayed equation
The first Stab(S) should be Stab(s).
Chapter 9
Page 206, Proof of Theorem 9.9
The first sentence says φ : G → Perm(S), but it is H that is acting on S, so it should instead read φ : H → Perm(S).
Page 211, first paragraph
The final sentence of the paragraph references Figure 9.6, when it should reference Figure 9.13.
Page 215, Proof of Theorem 9.12
The final paragraph could be clarified by changing the phrase "and thus it is not zero" to "and thus it is not zero mod p."
Page 219, Exercise 9.16
Although the word Ker in this exercise has extra spaces in it, it has the same meaning as elsewhere in the text, where it is typeset correctly. See also the erratum on page 292.
Page 220, Exercise 9.27
No reason is given for why ba must be some amb. Why could it not be amb2? You can prove that this cannot be the case in a number of different ways. Consider, for example, showing that <a> is a normal subgroup.
Chapter 10
Page 249, paragraph above 10.7.2
The following sentence is inaccurate.
Furthermore, no group containing A5 can be solvable, because the smallest first step in any chain of normal subgroups in such a group would be the invalid step {e} ⊲ A5.
It should be rewritten along the following lines.
Furthermore, no group containing A5 can be solvable. No step A ⊲ B in such a chain can have the quotient B/A isomorphic to A5, because it is not abelian. And yet it can also be shown that a step in the chain including just "part" of A5 could be used to reveal a normal subgroup in A5, an impossibility.
Page 255, Exercise 10.9
In part (e), you may also assume that √6 is irrational.
Page 258, Exercise 10.22
See the erratum for the solution to part (c), below. The same problem does not appear in any of the other three parts of this exercise, because in each case, arguments that are not too difficult can be made for the irreducibility of the resulting polynomial.
Pages 259-260, Exercise 10.29
The exercise claims to show the diagrams of finite fields of orders 5 and 8, but from the diagrams it is clear that this is a typo; they show the finite fields of orders 4 and 8. Thus when part (b) of the problem asks you to create the diagram for the finite field of order 4, this does not make sense since you were already given it. Consider instead creating the diagram for the finite field of order 5.
Appendix
Page 264, Figure A.2
The figure was incorrect in two ways; the right-hand friezes in the top square were incorrectly labeled, and the arced arrows along the right side were backwards. A corrected version appears here.
Page 273, Answer to Exercise 8.6
Part (b) is ambiguous. It asks if the set of elements to which φ maps K will be normal, but it does not say normal in what group. It will always be normal in Im(φ), but not always normal in H.
Page 273, Answer to Exercise 8.29
The Zn should be Cn instead.
Page 278, Answer to Exercise 9.12
It should instead state that Exercise 6.31 guides you through creating the counterexample that Exercise 9.12 requests.
Page 279, Answers to Exercises 9.22(c) and 9.27
The same correction regarding the order of factors in the semidirect product applies here as it did in earlier exercises. The product of C4 with C3 should be C3 with C4 instead, and the product of C3 with C7 should be C7 with C3 instead.
Page 282, Answer to Exercise 10.22(b)
The polynomial I create may be irreducible, but it is not obvious whether it is or not. I should have designed the exercise better; consider instead changing the 1 to a 2, so that r is the fifth root of 2+√2. Then the polynomial created is r10 - 4r5 + 2, which is irreducible by the Eisenstein Criterion (Theorem 10.4).
Index
Page 292, homomorphism kernel
Appears redundantly, due to typesetting error on page 219.