List of typos found in contemporary abstract algebra, 10th edition.

• p78 ch 4 example 4, instead of $\langle 1 \rangle = \langle 1 \rangle$, we should have $\langle 1 \rangle = \{ 1 \}$
• p131 Ch 6 Ex 6 In step 2, change {$MAM = MBA\inv$} to $MAM = MBM\inv$
• p132 Ch 6 Thm 6.1 Part 7, change {$G$ and $G$} to $G$ and $\overline{G}$
• p138 Ch 6 Thm 6.4 at the end with the $T(\alpha \beta)$ stuff, at the underbrace, change it to $\beta(1)$ times. Specifically $\underbrace{\alpha(1 + 1 + \cdots + 1)}_{\beta(1)}$ becomes $\underbrace{\alpha(1 + 1 + \cdots + 1)}_{\beta(1) \text{ times}}$ (Fixing the other one is left as an exercise)
• p165 ch 7 ex 32 On the last line, change "or order 2" to "of order 2".
• p187 ch 8 ex 39 On the first line, $G$ should be $\set{3^m 6^n \mid m, n \in Z}$ instead of $\set{3^m 6^m \mid m, n \in Z}$. Similar issue with the last line.
• p187 ch8 ex 40 There is a missing bracket. The corrected bit should look like $\abs{\paren{a_1, a_2, \dots, a_n}} = \infty$
• p221 ch 10 thm 10.1 On point 2, add in the missing brace. It should look like $\phi\paren{g^n} = \paren{\phi(g)}^n$ instead.
• p221 ch 10 thm 10.1 At the beginning of the proof of property 3, insert "Let $\overline e$ be the identity of $\overline G$.
• p221 ch 10 thm 10.1 At the line "imply that $e = \phi\paren e = \phi\paren{g^n} = \paren{\phi(g)}^n$", add an overline over the $e$ at the start to look like: \[ \overline e = \phi\paren e = \phi\paren{g^n} = \paren{\phi(g)}^n \]
• p223 ch 10 ex 8 \[ \operatorname{sgn}\paren\sigma = \begin{cases} +1 &\text{if $\sigma$ is an even permutation} \\ -1 &\text{if $\sigma$ is an even permutation} \end{cases} \] Change the second even to odd.
• p247 ch11 lemma 2 At the line "Since $b$ was chosen to have smallest order such that $b \in \gen a$", change it to $b \notin \gen a$.
• p247 ch 11 lemma 2 At the line "For if $x \in \gen a \cap K$, then $\overline x \in \gen{\overline a} \cap \overline K = \set{\overline e} = ${$\gen b$}", change $\gen b$ to $\{\gen b\}$
• p249 ch 11 lemma 4 At the line "Since ($n' = M'$)" change the $M'$ to $m'$
• p262 ch 12 ex 20 Change $R_0 \subset R_1 \subset R_2 \cdots$ to $R_0 \supset R_1 \supset R_2 \cdots$
• p274 ch 13 exercises Exercises 15 and 16 are duplicated (Chapter 12 Exercises 30 and 31) respectively
• p289 ch 14 ex 41 Change "communitive" to "commutative"
• p307 ex 50 If $R$ is a field prove the mapping is an isomorphism. Without the assumption that $R$ is finite, it is impossible. See Wikipedia and Math StackExchange
• p307 ex 53a Change $\phi -1 (A)$ to $\phi\inv(A)$
• p 320 ch 16 ex 14 Change "the converse is to not true" to "the converse is not true"
• p 321 ch 16 ex 29 Change "give an example of be a commutative ring" to "give an example of a commutative ring".
• p 344 ch 18 example 1 "so that $1 + \sqrt{-3}$ divides $2 \cdot 2$" add in the missing word "means": "so that means $1 + \sqrt{-3}$ divides $2 \cdot 2$"
• p 353 ch 18 example 4 Change $p_2$ to $p^2$. See Theorem 16.4 for Eisenstein's Criterion.
• p 367 ch 19 line 19 Change $-1 + \langle x^2 = 1 \rangle$ to $-1 + \langle x^2 + 1 \rangle$.

possible improvements

• p 326 example 3 "Since 1 is the only divisor of 40 that is congruent to 1 modulo 5"... Note that we only should be considering divisors of 40/5 by Sylow's 3rd Theorem.