\documentclass[numbers=enddot,12pt,final,onecolumn,notitlepage]{scrartcl}%
\usepackage{amsfonts}
\usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage}
\usepackage[all,cmtip]{xy}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{framed}
\usepackage{comment}
\usepackage{color}
\usepackage{hyperref}
\usepackage{ifthen}
\usepackage[sc]{mathpazo}
\usepackage[T1]{fontenc}
\usepackage{needspace}
\usepackage{tabls}
\usepackage{graphicx}%
\setcounter{MaxMatrixCols}{30}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.50.0.2960}
%TCIDATA{LastRevised=Saturday, September 26, 2026 23:51:33}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%BeginMSIPreambleData
\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%EndMSIPreambleData
\newcounter{exer}
\theoremstyle{definition}
\newtheorem{theo}{Theorem}[section]
\newenvironment{theorem}[1][]
{\begin{theo}[#1]\begin{leftbar}}
{\end{leftbar}\end{theo}}
\newtheorem{lem}[theo]{Lemma}
\newenvironment{lemma}[1][]
{\begin{lem}[#1]\begin{leftbar}}
{\end{leftbar}\end{lem}}
\newtheorem{prop}[theo]{Proposition}
\newenvironment{proposition}[1][]
{\begin{prop}[#1]\begin{leftbar}}
{\end{leftbar}\end{prop}}
\newtheorem{defi}[theo]{Definition}
\newenvironment{definition}[1][]
{\begin{defi}[#1]\begin{leftbar}}
{\end{leftbar}\end{defi}}
\newtheorem{remk}[theo]{Remark}
\newenvironment{remark}[1][]
{\begin{remk}[#1]\begin{leftbar}}
{\end{leftbar}\end{remk}}
\newtheorem{coro}[theo]{Corollary}
\newenvironment{corollary}[1][]
{\begin{coro}[#1]\begin{leftbar}}
{\end{leftbar}\end{coro}}
\newtheorem{conv}[theo]{Convention}
\newenvironment{convention}[1][]
{\begin{conv}[#1]\begin{leftbar}}
{\end{leftbar}\end{conv}}
\newtheorem{quest}[theo]{Question}
\newenvironment{question}[1][]
{\begin{quest}[#1]\begin{leftbar}}
{\end{leftbar}\end{quest}}
\newtheorem{warn}[theo]{Warning}
\newenvironment{warning}[1][]
{\begin{warn}[#1]\begin{leftbar}}
{\end{leftbar}\end{warn}}
\newtheorem{conj}[theo]{Conjecture}
\newenvironment{conjecture}[1][]
{\begin{conj}[#1]\begin{leftbar}}
{\end{leftbar}\end{conj}}
\newtheorem{exam}[theo]{Example}
\newenvironment{example}[1][]
{\begin{exam}[#1]\begin{leftbar}}
{\end{leftbar}\end{exam}}
\newtheorem{exmp}[exer]{Exercise}
\newenvironment{exercise}[1][]
{\begin{exmp}[#1]\begin{leftbar}}
{\end{leftbar}\end{exmp}}
\newenvironment{statement}{\begin{quote}}{\end{quote}}
\newenvironment{fineprint}{\begin{small}}{\end{small}}
\iffalse
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\newenvironment{convention}[1][Convention]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\newenvironment{question}[1][Question]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\fi
\let\sumnonlimits\sum
\let\prodnonlimits\prod
\let\cupnonlimits\bigcup
\let\capnonlimits\bigcap
\renewcommand{\sum}{\sumnonlimits\limits}
\renewcommand{\prod}{\prodnonlimits\limits}
\renewcommand{\bigcup}{\cupnonlimits\limits}
\renewcommand{\bigcap}{\capnonlimits\limits}
\setlength\tablinesep{3pt}
\setlength\arraylinesep{3pt}
\setlength\extrarulesep{3pt}
\setlength\textheight{22.5cm}
\setlength\textwidth{14.8cm}
\newenvironment{verlong}{}{}
\newenvironment{vershort}{}{}
\newenvironment{noncompile}{}{}
\excludecomment{verlong}
\includecomment{vershort}
\excludecomment{noncompile}
\newcommand{\id}{\operatorname{id}}
\newcommand{\rev}{\operatorname{rev}}
\newcommand{\conncomp}{\operatorname{conncomp}}
\newcommand{\conn}{\operatorname{conn}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\powset}[2][]{\ifthenelse{\equal{#2}{}}{\mathcal{P}\left(#1\right)}{\mathcal{P}_{#1}\left(#2\right)}}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\tup}[1]{\left( #1 \right)}
\newcommand{\ive}[1]{\left[ #1 \right]}
\newcommand{\verts}[1]{\operatorname{V}\left( #1 \right)}
\newcommand{\edges}[1]{\operatorname{E}\left( #1 \right)}
\newcommand{\arcs}[1]{\operatorname{A}\left( #1 \right)}
\newcommand{\underbrack}[2]{\underbrace{#1}_{\substack{#2}}}
\newcommand{\are}{\ar@{-}}
\newcommand{\arebi}[1][]{\ar@{<-}@/_/[#1] \ar@/^/[#1]}
\ihead{Math 331, Fall 2026: homework set 1}
\ohead{page \thepage}
\cfoot{}
\begin{document}

\begin{center}
\textbf{Math 331, Fall 2026: homework set 1}

due date: Monday 5 October 2026 at 3:30 PM on Canvas.

Please solve \textbf{5} of the 10 exercises!
\end{center}

\begin{exercise}
For the following five binary operations $\ast$ on the set $\mathbb{Z}$,
determine whether $\left(  \mathbb{Z},\ast\right)  $ is a semigroup, a monoid,
a group, or none of these: \renewcommand{\theenumi}{\alph{enumi}} \renewcommand{\labelenumi}{\textbf{(\theenumi)}}

\begin{enumerate}
\item the operation $\ast$ given by $a\ast b=\max\left\{  a,b\right\}  $;

\item the operation $\ast$ given by $a\ast b=a+b+1$;

\item the operation $\ast$ given by $a\ast b=\left\vert ab\right\vert $;

\item the operation $\ast$ given by $a\ast b=ab+a+b+1$;

\item the operation $\ast$ given by $a\ast b=ab-a-b+2$.
\end{enumerate}

Proofs are not required in this exercise.
\end{exercise}

If $s$ is an element of a monoid $\left(  S,\ast\right)  $, and if $n$ is a
nonnegative integer, then $s^{n}$ denotes the $n$-fold product
$\underbrace{s\ast s\ast\cdots\ast s}_{n\text{ times }s}$. For $n=0$, this is
simply the neutral element of $\left(  S,\ast\right)  $. If $s$ is furthermore
invertible, then $s^{n}$ is defined for negative integers $n$ as well, namely
by setting $s^{n}=\left(  s^{-1}\right)  ^{-n}$ when $n$ is negative.

\begin{exercise}
Let $\left(  S,\ast\right)  $ be a monoid with neutral element $e$. Let $s\in
S$. \renewcommand{\theenumi}{\alph{enumi}} \renewcommand{\labelenumi}{\textbf{(\theenumi)}}

\begin{enumerate}
\item Prove that if $s^{3}=e$ and $s^{5}=e$, then $s=e$.

\item Prove that if $s^{6}=e$ and $s^{10}=e$, then $s^{2}=e$.

\item Is it true that if $s^{6}=e$ and $s^{10}=e$, then $s=e$ ?
\end{enumerate}
\end{exercise}

\begin{exercise}
Let $\left(  S,\ast\right)  $ be a semigroup. Let $a,b\in S$ be such that $ab$
is central in $S$. Prove that%
\[
\left(  ab\right)  ^{n}=a^{n}b^{n}\ \ \ \ \ \ \ \ \ \ \text{for all positive
integers }n\text{.}%
\]

\end{exercise}

\begin{exercise}
Let $\left(  S,\ast\right)  $ be a monoid, and let $a\in S$ be an invertible
element. Let $b\in S$. \renewcommand{\theenumi}{\alph{enumi}} \renewcommand{\labelenumi}{\textbf{(\theenumi)}}

\begin{enumerate}
\item Prove that $\left(  aba^{-1}\right)  ^{k}=ab^{k}a^{-1}$ for each
nonnegative integer $k$.

\item Now assume that $b$, too, is invertible. Prove that $\left(
aba^{-1}\right)  ^{k}=ab^{k}a^{-1}$ holds for all integers $k$, including the
negative ones.
\end{enumerate}
\end{exercise}

\begin{exercise}
Consider a group $G=\left\{  e,a,b,c,d\right\}  $ with five elements, of which
you only know that $ee=e$ and $aa=b$ and $ab=c$ and $ac=d$. Reconstruct the
entire multiplication table of this group, i.e., compute all products $xy$
with $x,y\in G$.%
\[%
\begin{tabular}
[c]{|c||c|c|c|c|c|}\hline
$xy$ & $y=e$ & $y=a$ & $y=b$ & $y=c$ & $y=d$\\\hline\hline
$x=e$ & $e$ &  &  &  & \\\hline
$x=a$ &  & $b$ & $c$ & $d$ & \\\hline
$x=b$ &  &  &  &  & \\\hline
$x=c$ &  &  &  &  & \\\hline
$x=d$ &  &  &  &  & \\\hline
\end{tabular}
\]
Just fill in this table; don't show your proof. \medskip

[You should not assume that $e$ is the neutral element just because I called
it $e$. But it follows from $ee=e$ -- do you see how?]
\end{exercise}

\begin{exercise}
\label{exe.group-sudoku6}The following is the multiplication table of a group
$\left\{  e,p,q,r,u,v\right\}  $ with $6$ elements:%
\[%
\begin{tabular}
[c]{|c||c|c|c|c|c|c|}\hline
$xy$ & $y=e$ & $y=p$ & $y=q$ & $y=r$ & $y=u$ & $y=v$\\\hline\hline
$x=e$ & $e$ &  &  &  &  & \\\hline
$x=p$ &  & $e$ &  &  & $q$ & \\\hline
$x=q$ &  &  & $e$ &  &  & \\\hline
$x=r$ &  &  &  &  &  & \\\hline
$x=u$ &  & $r$ &  &  & $v$ & $e$\\\hline
$x=v$ &  &  &  &  &  & \\\hline
\end{tabular}
\]
Fill in the missing values. (Again, don't write down your argument.)
\end{exercise}

\begin{exercise}
Let $\left(  S,\ast\right)  $ be a nonempty semigroup. Assume that for each
$a\in S$, the maps%
\begin{align*}
L_{a}:S &  \rightarrow S,\\
x &  \mapsto ax
\end{align*}
and%
\begin{align*}
R_{a}:S &  \rightarrow S,\\
x &  \mapsto xa
\end{align*}
are bijective. Prove that $\left(  S,\ast\right)  $ is a group. \medskip

[\textbf{Hint:} How would you construct a neutral element?]
\end{exercise}

A \textbf{submonoid} of a monoid $\left(  S,\ast\right)  $ is defined to be a
subset $T$ of $S$ that contains the neutral element of $S$ and is closed under
the operation $\ast$ (that is, satisfies $a\ast b\in T$ for all $a,b\in T$).
For example, $\mathbb{N}$ is a submonoid of $\left(  \mathbb{Z},+\right)  $.
Note that a submonoid $T$ of a monoid $\left(  S,\ast\right)  $ always becomes
a monoid itself if we restrict the operation $\ast$ to it (more precisely: to
$T\times T$).

\begin{exercise}
Let $\mathbb{Z}_{\operatorname*{ev}}$ be the set of all even integers, and let
$\mathbb{Z}_{\operatorname*{od}}$ be the set of all odd integers. Consider the
two monoids $\left(  \mathbb{Z},+\right)  $ and $\left(  \mathbb{Z}%
,\cdot\right)  $. \renewcommand{\theenumi}{\alph{enumi}} \renewcommand{\labelenumi}{\textbf{(\theenumi)}}

\begin{enumerate}
\item Is $\mathbb{Z}_{\operatorname*{ev}}$ a submonoid of $\left(
\mathbb{Z},+\right)  $ ?

\item Is $\mathbb{Z}_{\operatorname*{od}}$ a submonoid of $\left(
\mathbb{Z},+\right)  $ ?

\item Is $\mathbb{Z}_{\operatorname*{ev}}$ a submonoid of $\left(
\mathbb{Z},\cdot\right)  $ ?

\item Is $\mathbb{Z}_{\operatorname*{od}}$ a submonoid of $\left(
\mathbb{Z},\cdot\right)  $ ?

\item Is $\left\{  1\right\}  \cup\mathbb{Z}_{\operatorname*{ev}}$ a submonoid
of $\left(  \mathbb{Z},\cdot\right)  $ ?
\end{enumerate}
\end{exercise}

\begin{exercise}
Let $\left(  S,\ast\right)  $ be a monoid with neutral element $e$. Let $a\in
S$ be any element. The \textbf{centralizer} of $a$ is defined to be the set
$C_{a}$ of all $s\in S$ that commute with $a$ (that is, that satisfy $as=sa$).
\renewcommand{\theenumi}{\alph{enumi}} \renewcommand{\labelenumi}{\textbf{(\theenumi)}}

\begin{enumerate}
\item Prove that $C_{a}$ is a submonoid of $S$ (that is: $e\in C_{a}$ and
$st\in C_{a}$ for all $s,t\in C_{a}$).

\item Assume that $S$ is a group. Prove that $C_{a}$ is a subgroup of $S$
(that is: in addition to what was shown in part \textbf{(a)}, we also have
$s^{-1}\in C_{a}$ for each $s\in C_{a}$).

\item What is $C_{e}$?
\end{enumerate}
\end{exercise}

Recall that a semigroup $\left(  S,\ast\right)  $ is said to be
\textbf{abelian} if $ab=ba$ for all $a,b\in S$.

\begin{exercise}
\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{\textbf{(\theenumi)}} Let $\left(  S,\ast\right)  $
be a monoid with neutral element $e$. Let $n$ be a positive integer. Define
the set%
\[
T_{n}:=\left\{  s\in S\ \mid\ s^{n}=e\right\}  .
\]


\begin{enumerate}
\item Prove that if $\left(  S,\ast\right)  $ is abelian, then $T_{n}$ is a
submonoid of $S$.

\item Prove that this does not necessarily hold when $\left(  S,\ast\right)  $
is not abelian.

[\textbf{Hint:} Try the group from Exercise \ref{exe.group-sudoku6} and $n=2$.]
\end{enumerate}
\end{exercise}


\end{document}