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\section*{Math 403/503 Spring 2024}
\subsection*{Homework 1}
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\item Let $(M,\cdot,e)$ be a monoid. Show that $M^\times$ is a submonoid of $M$, and that $M^\times$ is a group (using the same operation as in $M$).
\item Determine for which sets $X$ the monoid $(X^X,\circ,\mathrm{Id}_X)$ is commutative.
\item Let $(M,\cdot,e)$ and $(N,\ast,e')$ be monoids. Show that $M\times N$ becomes a monoid in a natural way.
\item Let $G=S_3$ and $H=\{(1), (12)\}$, $N=\{(1),(123),(132)\}$.
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\item Is it true that $gH=Hg$ for all $g\in G$?
\item Is it true that $gN=Ng$ for all $g\in G$?  (Recall that $gH=\{gh\mid h\in H\}$ and similarly for $Hg$.)
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