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\ihead{Math 331, Fall 2026: homework set 2}
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\begin{document}

\begin{center}
\textbf{Math 331, Fall 2026: homework set 2}

due date: Wednesday 14 October 2026 at 3:30 PM on Canvas.

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

As discussed in \S 1.14, we write all semigroups (including monoids and
groups) multiplicatively unless we say otherwise.

\begin{exercise}
Let $S$ be a semigroup, written multiplicatively. Let $a$ and $b$ be two
elements of $S$ that commute.

\begin{enumerate}
[label=\textbf{(\alph*)},ref=\alph*,leftmargin=*,topsep=4pt,partopsep=0pt,itemsep=4pt,parsep=0pt]

\item Prove that $a^{i}b=ba^{i}$ for each positive integer $i$.

\item Prove that $a^{i}b^{j}=b^{j}a^{i}$ for all positive integers $i$ and $j$.

\item Prove that $\left(  ab\right)  ^{n}=a^{n}b^{n}$ for all positive
integers $n$.
\end{enumerate}
\end{exercise}

\begin{exercise}
Let $a$ be an invertible element of a monoid $M$. Prove that $o\left(
a^{-1}\right)  =o\left(  a\right)  $. (In particular, this means that
$o\left(  a\right)  $ and $o\left(  a^{-1}\right)  $ are either both finite or
both infinite.)
\end{exercise}

\begin{exercise}
Find a nonabelian monoid of size $3$.

[\textbf{Hint:} Start with a nonabelian semigroup of size $2$.]
\end{exercise}

\begin{exercise}
Let $S$ be a semigroup. Prove that we can obtain a monoid $S\cup\left\{
1\right\}  $ from $S$ by adjoining a new element -- let us call it $1$ -- and
setting $s\ast1:=s$ and $1\ast s:=s$ and $1\ast1:=1$ for all $s\in S$. (This
is called \textbf{adjoining a neutral element} to $S$. Note that if $S$
already had a neutral element $1_{\operatorname*{old}}$, then the new $1$ will
dethrone $1_{\operatorname*{old}}$, since $1\ast1_{\operatorname*{old}}$ is no
longer $1$.)
\end{exercise}

\begin{exercise}
Let $S$ be a monoid. Let $a\in S$ have an inverse. Prove that%
\[
\left(  a^{n}\right)  ^{m}=a^{nm}\ \ \ \ \ \ \ \ \ \ \text{for all integers
}n\text{ and }m.
\]


[\textbf{Hint:} Make sure your argument allows any or both of $n$ and $m$ to
be negative.]
\end{exercise}

\begin{exercise}
Let $G$ be a group. Prove that $G$ is abelian if and only if all $a,b\in G$
satisfy $\left(  ab\right)  ^{-1}=a^{-1}b^{-1}$.
\end{exercise}

\begin{exercise}
Let $S$ be a monoid, written multiplicatively. Let $a$ and $b$ be two elements
of $S$ that commute and have inverses.

\begin{enumerate}
[label=\textbf{(\alph*)},ref=\alph*,leftmargin=*,topsep=4pt,partopsep=0pt,itemsep=4pt,parsep=0pt]

\item Prove that $a^{i}b=ba^{i}$ for each integer $i$.

\item Prove that $a^{i}b^{j}=b^{j}a^{i}$ for all integers $i$ and $j$.

\item Prove that $\left(  ab\right)  ^{n}=a^{n}b^{n}$ for all integers $n$.
\end{enumerate}

[\textbf{Hint:} You are allowed to use all preceding exercises even if you
didn't solve them.]
\end{exercise}

\begin{exercise}
Let $G$ be a group. Let $u,v\in G$. Prove that $o\left(  uv\right)  =o\left(
vu\right)  $.
\end{exercise}

\begin{exercise}
Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be the map that sends each
$n\in\mathbb{N}$ to $n+1$. (Recall that $\mathbb{N}=\left\{  0,1,2,\ldots
\right\}  $.)

Let $g:\mathbb{N}\rightarrow\mathbb{N}$ be the map that sends each
$n\in\mathbb{N}$ to $\max\left\{  n-1,0\right\}  $ (that is, it sends
$0,1,2,3,4,\ldots$ to $0,0,1,2,3,\ldots$, respectively).

Prove that $g\circ f=\operatorname*{id}$ but $f\circ g\neq\operatorname*{id}$.
Conclude that the preceding exercise does not generalize to monoids. (What is
the monoid here?)
\end{exercise}

\begin{exercise}
Consider the two permutations $u=t_{1,2}t_{3,4}t_{5,6}$ and $v=t_{2,3}t_{4,5}$
in the symmetric group $S_{6}=S_{\left\{  1,2,3,4,5,6\right\}  }$. Here,
$t_{i,j}$ denotes the transposition that swaps two given elements $i$ and $j$.

\begin{enumerate}
[label=\textbf{(\alph*)},ref=\alph*,leftmargin=*,topsep=4pt,partopsep=0pt,itemsep=4pt,parsep=0pt]

\item Compute $uv$.

\item Compute $o\left(  u\right)  $ and $o\left(  v\right)  $ and $o\left(
uv\right)  $.
\end{enumerate}
\end{exercise}


\end{document}