\documentclass[numbers=enddot,12pt,final,onecolumn,notitlepage]{scrartcl}%
\usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{color}
\usepackage{hyperref}
\usepackage[sc]{mathpazo}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{enumitem}
\usepackage{graphicx}%
\setcounter{MaxMatrixCols}{30}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.50.0.2960}
%TCIDATA{LastRevised=Tuesday, July 28, 2026 22:15:37}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%BeginMSIPreambleData
\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%EndMSIPreambleData
\setlength\textheight{22.5cm}
\setlength\textwidth{15cm}
\setlength\oddsidemargin{0.5cm}
\setlength\evensidemargin{0.5cm}
\setlength\topmargin{-1cm}
\clearpairofpagestyles
\ihead{Errata to \textit{Introduction to Abstract Algebra}}
\ohead{\today}
\cfoot{\pagemark}
\newcommand{\booktitle}{\textit{Introduction to Abstract Algebra}}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\QQ}{\mathbb Q}
\newcommand{\RR}{\mathbb R}
\newcommand{\CC}{\mathbb C}
\newcommand{\NN}{\mathbb N}
\newcommand{\id}{\operatorname{id}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\lcm}{\operatorname{lcm}}
\begin{document}

\begin{center}
\textbf{Introduction to Abstract Algebra}

\textit{Samir Siksek}

MA136; PDF version dated September 22, 2020

\url{https://samirsiksek.github.io/siksek.github.io/teaching/aa/aanotes.pdf}

\textbf{Errata and comments} by Darij Grinberg
\end{center}

\noindent The page numbers below refer to the printed page numbers in the PDF
version dated September 22, 2020. This list is not claimed to be exhaustive.

All the errata below were found by GPT-5.6 Sol, then verified (and
occasionally reworded) by myself.

\appendix


\section{Corrections and comments}

\begin{enumerate}
\item \textbf{Page 2, last sentence of Section I.3:} Insert \textquotedblleft
be\textquotedblright\ in \textquotedblleft where you may
asked\textquotedblright.

\item \textbf{Page 4, item (2):} Insert \textquotedblleft a\textquotedblright%
\ in \textquotedblleft It isn't possible to write $\mathbb{R}$ in
straightforward way\textquotedblright.

\item \textbf{Page 5, Section II.3:} Replace \textquotedblleft If
$\mathbf{x},\mathbf{y}\in\mathbb{R}^{3}$ the $\mathbf{x}\times\mathbf{y}$ is
again in $\mathbb{R}^{3}$\textquotedblright\ by \textquotedblleft If
$\mathbf{x},\mathbf{y}\in\mathbb{R}^{3}$, then $\mathbf{x}\times\mathbf{y}$ is
again in $\mathbb{R}^{3}$\textquotedblright.

\item \textbf{Page 6, paragraph before Lemma II.2:} Insert \textquotedblleft
is\textquotedblright\ in \textquotedblleft but it one of the most important
results\textquotedblright.

\item \textbf{Page 7, first paragraph of Section II.6:} Replace
\textquotedblleft If the $S$ is finite\textquotedblright\ by \textquotedblleft
If $S$ is finite\textquotedblright.

\item \textbf{Page 7, Example II.3:} Delete the comma in \textquotedblleft The
result of the composition $a\circ b$, is found\textquotedblright.

\item \textbf{Page 16, paragraph before (III.6):} Replace \textquotedblleft
This pair of equations can re-expressed\textquotedblright\ by
\textquotedblleft This pair of equations can be re-expressed\textquotedblright.

\item \textbf{Page 17, first paragraph:} Replace ``we expect that matrix
equations (III.6) and (III.7) to be consistent'' by either ``we expect matrix
equations (III.6) and (III.7) to be consistent'' or ``we expect that matrix
equations (III.6) and (III.7) are consistent''.

\item \textbf{Page 18, Example III.8:} Replace ``$C$ represent projection onto
the $y$-axis'' by ``$C$ represents projection onto the $y$-axis''.

\item \textbf{Page 19, Subsection III.4.1:} Replace ``slight-of-hand'' by
``sleight of hand''.

\item \textbf{Page 19, proof of Theorem III.9:} Associativity of composition
shows that the two sides induce the same function, but the proof has not
explained why equality of the induced functions implies equality of the
matrices. It also uses, without stating it, the compatibility between matrix
multiplication and composition.

A short completion is as follows. For compatible matrices $X$ and $Y$, the
definition of matrix multiplication gives
\[
T_{XY}=T_{X}\circ T_{Y}.
\]
Hence $T_{(AB)C}=T_{A(BC)}$. If two $m\times q$ matrices $D$ and $E$ induce
the same function $\mathbb{R}^{q}\rightarrow\mathbb{R}^{m}$, then
$D\mathbf{e}_{j}=E\mathbf{e}_{j}$ for every standard basis vector
$\mathbf{e}_{j}$; but these vectors are the $j$-th columns of $D$ and $E$.
Thus $D=E$, and therefore $(AB)C=A(BC)$.

\item \textbf{Page 22, first sentence:} Replace the full stop in ``Should
there be such an inverse $A^{-1}$ for every $A$.'' by a question mark.

\item \textbf{Page 23, end of Example III.10:} In
\[
C^{-1}C=%
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}%
\begin{pmatrix}
0 & 0\\
0 & 1
\end{pmatrix}
=%
\begin{pmatrix}
0 & b\\
0 & d
\end{pmatrix}
,
\]
it is the \emph{top-left} entries that do not match those of $I_{2}$; the
bottom-left entries are both $0$.

\item \textbf{Page 24, Theorem III.11 and the surrounding discussion:} This is
true over a field such as $\mathbb{R}$, $\mathbb{Q}$, or $\mathbb{C}$. It is
false for invertibility inside $M_{2\times2}(\mathbb{Z})$: there the
determinant must be $\pm1$. This distinction becomes important in Exercise
VI.2 and Example XIV.25.

\item \textbf{Page 25, proof of Theorem III.12 (c):} Replace ``applying (ii)''
by ``applying (b)''.

\item \textbf{Page 26, derivation of the rotation matrix:} After
\textquotedblleft and let the angle $\overrightarrow{OP}$ makes with the
positive $x$-axis be $\phi$\textquotedblright, add \textquotedblleft(or pick
$\phi$ arbitrarily when $P=O$)\textquotedblright.

\item \textbf{Page 30, first bullet point:} Replace \textquotedblleft Like in
$\mathbb{R}^{2}$ we have addition and scalar multiplication\textquotedblright%
\ by \textquotedblleft Likewise, in $\mathbb{R}^{2}$ we have addition and
scalar multiplication\textquotedblright.

\item \textbf{Page 31, paragraph before the composition table:} Replace
\textquotedblleft This has the same effect at $\sigma_{1}$\textquotedblright%
\ by \textquotedblleft This has the same effect as $\sigma_{1}$%
\textquotedblright.

\item \textbf{Page 32, last paragraph:} Replace \textquotedblleft$\circ$
simply becomes composition of functions from $D_{4}$ to
itself\textquotedblright\ by \textquotedblleft$\circ$ simply becomes
composition of functions from $\{1,2,3,4\}$ to itself\textquotedblright.

\item \textbf{Page 33, Subsection IV.4.1:} Replace \textquotedblleft seeing
the $(R,\circ)$ is abelian\textquotedblright\ by \textquotedblleft seeing that
$(R,\circ)$ is abelian\textquotedblright.

\item \textbf{Page 34, Exercise IV.14:} Replace ``isoceles'' by ``isosceles''.

\item \textbf{Page 39, Theorem VI.1:} Insert \textquotedblleft
a\textquotedblright\ in \textquotedblleft$\operatorname{GL}_{2}(\mathbb{R})$
is group under multiplication of matrices\textquotedblright.

\item \textbf{Page 40, Section VI.2:} Replace \textquotedblleft In
otherwords\textquotedblright\ by \textquotedblleft In other
words\textquotedblright\ (unless an otherworld is being referred to, which
requires a different correction). Later in the same paragraph,
\textquotedblleft addition and multiplication is defined\textquotedblright%
\ should be \textquotedblleft addition and multiplication are
defined\textquotedblright.

\item \textbf{Page 41, proof of Theorem VI.5:} Insert \textquotedblleft
is\textquotedblright\ in \textquotedblleft To show that $\mathbb{Z}%
/m\mathbb{Z}$ a group\textquotedblright.

\item \textbf{Page 43, Example VII.1:} Replace ``The order of $\rho_{1}$ is
$D_{4}$ is $4$'' by ``The order of $\rho_{1}$ in $D_{4}$ is $4$''.

\item \textbf{Page 43, Example VII.2:} Replace the full stop after ``What are
the elements of finite order in $\mathbb{R}^{*}$'' by a question mark.

\item \textbf{Page 45, paragraph before Example VII.9:} Replace ``If there is
no such positive integer that $a$ has infinite order'' by ``If there is no
such positive integer, then $a$ has infinite order''.

\item \textbf{Page 45, Example VII.9:} Replace the full stop in ``How do we
know this.'' by a question mark.

\item \textbf{Page 47, first paragraph after the heading:} Insert ``be'' in
``we want $H$ to be a group''.

\item \textbf{Page 49, Example VIII.10:} Insert \textquotedblleft
a\textquotedblright\ in \textquotedblleft Hence $W$ is not subgroup of
$\mathbb{R}^{2}$\textquotedblright.

\item \textbf{Page 52, first sentence after Figure VIII.5:} Replace
\textquotedblleft Let's check that this is indeed a subgroup of $V_{\Pi}%
$\textquotedblright\ by \textquotedblleft Let's check that $V_{\Pi}$ is indeed
a subgroup of $\mathbb{R}^{3}$\textquotedblright.

\item \textbf{Page 53, end of Example VIII.17:} The question \textquotedblleft
Can you find a finite subgroup of $\mathbb{C}^{\ast}$ that isn't a subgroup of
$\mathbb{S}$?\textquotedblright\ is never explicitly answered in the notes.
Unless it is intended as a cliffhanger, it would be good to return to it (the
answer is \textquotedblleft no\textquotedblright; indeed, if $\alpha$ lies in
a finite subgroup of $\mathbb{C}^{\ast}$, then $\alpha^{n}=1$ for some $n>0$,
so $|\alpha|^{n}=1$ and hence $|\alpha|=1$).

\item \textbf{Page 54, proof of Theorem VIII.5:} Insert \textquotedblleft
is\textquotedblright\ in \textquotedblleft It usual when proving an `if and
only if' statement \ldots\textquotedblright.

\item \textbf{Page 56, opening paragraph:} Insert \textquotedblleft
a\textquotedblright\ in \textquotedblleft This is group where the operation is
multiplication of matrices\textquotedblright.

\item \textbf{Page 56, paragraph before the definition of $\operatorname{SO}%
_{2}(\mathbb{R})$:} Insert \textquotedblleft we\textquotedblright\ in
\textquotedblleft if compose two rotations\textquotedblright. In footnote 1,
replace \textquotedblleft a subgroup of a something\textquotedblright\ by
\textquotedblleft a subgroup of something\textquotedblright.

\item \textbf{Page 58, Example VIII.30:} Replace ``Examples of a non-trivial
subgroups'' by ``Examples of non-trivial subgroups''.

\item \textbf{Page 59, Example VIII.34:} Insert ``us'' in ``Lagrange's Theorem
tells us that the order of $U_{2}$ must divide the order of $U_{4}$''.

\item \textbf{Page 62, Example IX.3:} Replace \textquotedblleft$\mathbb{Z}%
/m\mathbb{Z}$\textquotedblright\ by \textquotedblleft$\mathbb{Z}/6\mathbb{Z}%
$\textquotedblright\ twice in this example, since the displayed table is
specifically for the six residue classes modulo $6$.

\item \textbf{Page 64, Theorem IX.12:} Insert \textquotedblleft
of\textquotedblright\ in \textquotedblleft The order $g$ divides the order of
$G$\textquotedblright.

\item \textbf{Page 68, Exercise X.3:} Replace \textquotedblleft$\{1,Z,\ldots
,Z^{5}\}$\textquotedblright\ by \textquotedblleft$\{I_{2},Z,\ldots,Z^{5}%
\}$\textquotedblright. The identity in this matrix group is $I_{2}$, not the
scalar $1$.

\item \textbf{Page 68, Exercise X.4:} Replace ``cylic'' by ``cyclic''.

\item \textbf{Page 70, Exercise XI.4:} Replace ``In otherwords'' by ``In other words''.

\item \textbf{Page 71, Example XI.7:} Replace ``One convenient way of a
specifying a line'' by ``One convenient way of specifying a line''.

\item \textbf{Page 72, Example XI.7:} Insert \textquotedblleft
is\textquotedblright\ in \textquotedblleft which in $L$\textquotedblright.
More importantly, replace \textquotedblleft$-\mathbf{v}_{1}=(-t_{1}%
)\mathbf{v}_{1}$\textquotedblright\ by \textquotedblleft$-\mathbf{v}%
_{1}=(-t_{1})\mathbf{v}$\textquotedblright, since $\mathbf{v}_{1}%
=t_{1}\mathbf{v}$.

\item \textbf{Page 73, Example XI.9:} Insert \textquotedblleft
the\textquotedblright\ in \textquotedblleft one of things you'll look
at\textquotedblright.

\item \textbf{Page 75, proof of Lemma XI.15:} Replace \textquotedblleft In
otherwords\textquotedblright\ by \textquotedblleft In other
words\textquotedblright. Later, replace \textquotedblleft By we \textit{can}
say multiply both sides\textquotedblright\ by \textquotedblleft But we
\textit{can} multiply both sides\textquotedblright.

\item \textbf{Page 76, Example XI.18:} Insert \textquotedblleft
at\textquotedblright\ in \textquotedblleft There we looked the cosets of the
circle subgroup\textquotedblright.

\item \textbf{Page 77, last line:} Replace ``ecstacy'' by ``ecstasy''.

\item \textbf{Page 80, Example XII.2:} Insert \textquotedblleft
the\textquotedblright\ in \textquotedblleft the small fractional part of
number\textquotedblright.

\item \textbf{Page 80, Example XII.4:} Insert \textquotedblleft
if\textquotedblright\ in \textquotedblleft if and only their constant terms
are equal\textquotedblright. In footnote 1, replace \textquotedblleft writing
$f$ and $g$\textquotedblright\ by \textquotedblleft writing $g$ and
$h$\textquotedblright, since these are the two polynomials in the displayed formula.

\item \textbf{Page 81, Exercise XII.5:} Insert \textquotedblleft
be\textquotedblright\ in \textquotedblleft for $a$ and $b$ to congruent modulo
$\{0\}$\textquotedblright.

\item \textbf{Page 81, Lemma XII.7:} Insert \textquotedblleft
and\textquotedblright\ after \textquotedblleft Let $a\in G$\textquotedblright.

\item \textbf{Page 81, definition of the quotient group:} Replace ``We define
the quotient group $(G/H,+)$ to the set'' by ``We define the quotient group
$(G/H,+)$ to be the set''.

\item \textbf{Page 82, Section XII.3:} Delete one occurrence of ``and'' in
``and and simplify by subtracting an integer''.

\item \textbf{Page 85, Lemma XII.16:} Replace \textquotedblleft Let
$a,a^{\prime},b,b^{\prime}$ satisfying\textquotedblright\ by \textquotedblleft
Let $a,a^{\prime},b,b^{\prime}$ satisfy\textquotedblright. In the following
paragraph, replace \textquotedblleft representatitives\textquotedblright\ by
\textquotedblleft representatives\textquotedblright\ and insert
\textquotedblleft is\textquotedblright\ in \textquotedblleft even though it
defined in terms of those representatives\textquotedblright.

\item \textbf{Page 85, paragraph before Lemma XII.17:} Delete the first
occurrence of ``addition'' in ``Are we sure addition that addition is
well-defined \ldots''.

\item \textbf{Page 88, definition of injections and surjections:} It is safer
not to say that $B$ is \textquotedblleft the \textit{range} of $f$%
\textquotedblright. Many texts reserve the word \textquotedblleft%
\emph{range}\textquotedblright\ for the image
\[
f(A)=\{f(a):a\in A\},
\]
which may be a proper subset of the codomain $B$. This distinction is
particularly useful when defining surjectivity.

\item \textbf{Pages 88--89, discussion of $f_{2}$:} Replace \textquotedblleft%
$f(1)\neq b$ and $f(2)\neq b$\textquotedblright\ by \textquotedblleft%
$f_{2}(1)\neq b$ and $f_{2}(2)\neq b$\textquotedblright.

\item \textbf{Page 91, Example XIII.11:} Replace $f_{2}^{n}\neq
\operatorname{id}_{A}$ by $f_{2}^{n}\neq\operatorname{id}_{\mathbb{Z}}$ (or
define $A=\mathbb{Z}$).

\item \textbf{Page 94, Subsection XIII.4.1:} Replace \textquotedblleft One way
of seeing this is convince ourselves\textquotedblright\ by \textquotedblleft
One way of seeing this is to convince ourselves\textquotedblright. Later,
replace \textquotedblleft the permutation $\{1,2,3\}$\textquotedblright\ by
\textquotedblleft the permutation of $\{1,2,3\}$\textquotedblright.

\item \textbf{Page 94, first paragraph of Section XIII.5:} In
\textquotedblleft It is not all obvious from this formula\textquotedblright,
replace \textquotedblleft all\textquotedblright\ by \textquotedblleft at
all\textquotedblright.

\item \textbf{Page 96, paragraph before Theorem XIII.23:} Delete one
occurrence of \textquotedblleft and\textquotedblright\ in \textquotedblleft
more concise and and more transparent\textquotedblright, and replace
\textquotedblleft surprize\textquotedblright\ by \textquotedblleft
surprise\textquotedblright.

\item \textbf{Page 96, Example XIII.24:} Replace \textquotedblleft are
repeatedly apply\textquotedblright\ by \textquotedblleft and repeatedly
apply\textquotedblright.

\item \textbf{Page 98, proof of Lemma XIII.29:} Replace \textquotedblleft Let
$\sigma$ and $\tau$ be disjoint cycle\textquotedblright\ by \textquotedblleft
Let $\sigma$ and $\tau$ be disjoint cycles\textquotedblright.

\item \textbf{Pages 98--99, proof of Theorem XIII.23:} When arguing that
\textquotedblleft This contradicts our assumption that $a$ does not appear in
the list $1,\rho1,\ldots,\rho^{u-1}1$\textquotedblright, you are tacitly using
the fact that $\rho^{k}1$ does appear in this list (because $\rho^{u}1=1$
ensures that all of $1,\rho1,\rho^{2}1,\ldots$ appear in this list). Perhaps
it would be good to be less tacit about it.

\item \textbf{Pages 100 and 103, the five-transposition decomposition of
$(1,2,3,4)$:} The displayed decomposition uses the symbol $5$, so it is an
identity in $S_{n}$ only for $n\geq5$. State explicitly that $(1,2,3,4)$ is
being viewed as an element of $S_{5}$ (or replace the example by a
five-transposition decomposition using only $1,2,3,4$).

\item \textbf{Page 106, Exercise XIII.51:} The puzzle promoter's name is
\textbf{Sam Loyd}, not ``Sam Lloyd''.

\item \textbf{Page 107, property (iv) in the definition of a ring:} Insert
\textquotedblleft is\textquotedblright\ in \textquotedblleft there an element,
denoted by $-a$\textquotedblright.

\item \textbf{Page 108, paragraph before Section XIV.2:} Replace ``Observe,
from properties (i)--(v), if \ldots'' by ``Observe from properties (i)--(v)
that, if \ldots''.

\item \textbf{Page 109, Example XIV.6:} The argument uses $a\cdot0=0$ without
having proved this ring identity. One can insert
\[
a\cdot0=a(0+0)=a\cdot0+a\cdot0;
\]
cancelling $a\cdot0$ in the additive group gives $a\cdot0=0$. Similarly,
$0\cdot a=0$. The printed conclusion $a=a\cdot1=a\cdot0=0$ then follows.

It is probably best to state the identities $a\cdot0=0$ and $0\cdot a=0$ as a
lemma before or after this example, since their usefulness is not limited to
trivial rings.

\item \textbf{Page 110, paragraph after the definition of a subring:} Insert
\textquotedblleft be\textquotedblright\ in \textquotedblleft we want $S$ to a
ring\textquotedblright, and replace \textquotedblleft\textit{the same two
binary operations} that makes $R$ a ring\textquotedblright\ by
\textquotedblleft the same two binary operations that make $R$ a
ring\textquotedblright.

\item \textbf{Page 111, Example XIV.13:} With
\[
\alpha=a_{1}+a_{2}i,\qquad\beta=b_{1}+b_{2}i,
\]
the product should be
\[
\alpha\beta=(a_{1}b_{1}-a_{2}b_{2})+(a_{1}b_{2}+a_{2}b_{1})i,
\]
not
\[
(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+a_{2}b_{1})i.
\]
Consequently, the subsequent list of integers should be
\[
a_{1}+b_{1},\quad a_{2}+b_{2},\quad-a_{1},\quad-a_{2},\quad a_{1}b_{1}%
-a_{2}b_{2},\quad a_{1}b_{2}+a_{2}b_{1}.
\]
The printed list has $-b_{1}$ in place of $-a_{2}$ and repeats the erroneous
real part of the product.

\item \textbf{Page 112, continuation of Example XIV.15:} In the sentence
beginning ``Clearly $-\alpha,\alpha\beta$ are in $S$'', replace
\[
\text{``$-a,a+b,r,r+s$ are integers''} \quad\text{by}\quad
\text{``$-a,ab,r,r+s$ are integers''.}
\]


\item \textbf{Page 112, first paragraph of Section XIV.4:} Replace
\textquotedblleft we defined $\mathbb{R}^{\ast},\mathbb{Q}^{\ast}%
,\mathbb{C}^{\ast}$ be removing\textquotedblright\ by \textquotedblleft we
defined $\mathbb{R}^{\ast},\mathbb{Q}^{\ast},\mathbb{C}^{\ast}$ by
removing\textquotedblright, and insert \textquotedblleft a\textquotedblright%
\ in \textquotedblleft We found that $\mathbb{R}^{\ast}$ is
group\textquotedblright.

\item \textbf{Page 112, definition of a unit:} Replace ``An element $u$ is
called a unit'' by ``An element $u\in R$ is called a unit of $R$''. This makes
the ambient ring explicit before the inverse $v\in R$ is introduced.

\item \textbf{Page 114, Example XIV.25:} Replace ``Taking determinants, are
recalling'' by ``Taking determinants, and recalling''.

\item \textbf{Page 115, Example XIV.26:} Insert \textquotedblleft
a\textquotedblright\ in \textquotedblleft To be unit it is not enough for this
matrix to be invertible\textquotedblright. (Also perhaps add a comma after
\textquotedblleft unit\textquotedblright.)

\item \textbf{Page 119, definition of a field:} This definition should also
require that $1\neq0$ in $F$ (so that the zero ring is not a field).

\item \textbf{Page 119, Example XV.4:} Replace \textquotedblleft We want to
show that existence of some $\beta\in\mathbb{Q}[i]$\textquotedblright\ by
either \textquotedblleft We want to show the existence of some $\beta
\in\mathbb{Q}[i]$\textquotedblright\ or \textquotedblleft We want to show that
there exists some $\beta\in\mathbb{Q}[i]$\textquotedblright.

\item \textbf{Page 121, Example XVI.1:} Insert \textquotedblleft
to\textquotedblright\ in \textquotedblleft You don't have be very clever
here!\textquotedblright.

\item \textbf{Page 121, Example XVI.4:} Insert \textquotedblleft
time\textquotedblright\ (or remove \textquotedblleft a\textquotedblright) in
\textquotedblleft It would take us a very long to run through the elements
\ldots\textquotedblright.

\item \textbf{Page 123, definition of Euler's $\phi$-function:} The notes
previously introduced $\mathbb{Z}/m\mathbb{Z}$ only for $m\geq2$, whereas this
definition begins with $m\geq1$. Either change the hypothesis here to $m\geq
2$, or extend the definition of $\mathbb{Z}/m\mathbb{Z}$ to $m=1$ (this does
not require any changes) and explain that its one-element unit group gives
$\phi(1)=1$.
\end{enumerate}


\end{document}