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\begin{center}
{\large MATH 108 Winter 2019 \ -  \ {\bf Problem Set 8}}

% Put your name and section here.
{\large due March 15}
\end{center}


\begin{enumerate}
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	\item \label{GH}
	\begin{enumerate}
	 \item Given that $G = \{e,u,v,w\}$ is a group of order 4 with identity $e$, $u^2 = v$ and $v^2 = e$, construct the operation table for $G$.
	 \item Given that $H = \{a,b,c,d\}$ is a group of order 4 with identity $a$ and $b^2 = c^2 = d^2 = a$, construct the operation table for $H$.
	\end{enumerate}

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	\item Find all subgroups of the symmetric group on three elements, $\mathfrak{S}_3$.

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	\item Let $G$ be the symmetry group of a square.  Let $e \in G$ be the identity element. Let $r \in G$ denote a $90^\circ$ counter-clockwise rotation of the square.  Let $s \in G$ denote a reflection of the square across a vertical line through the center.  List the eight elements of $G$ in terms of $r$ and $s$ and find the order of each element.  (You can physically model $G$ by rotating and flipping a square of paper.)

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	\item Let $f:\RR \to \RR$ be defined by $f(x) = x^3$.
	\begin{enumerate}
	 \item Is $f:(\RR,+) \to (\RR,+)$ a group homomorphism?  Justify your answer.
	 \item Is $f:(\RR\setminus\{0\},\cdot) \to (\RR\setminus\{0\},\cdot)$ a group homomorphism?  Justify your answer.
	\end{enumerate}

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	\item Let $G$ be a group (represented multiplicatively) and $H$ a subgroup of $G$.  Define a relation $\sim$ on $G$ by $a \sim b$ if and only if $ab^{-1} \in H$.
	\begin{enumerate}
	 \item Prove that $\sim$ is an equivalence relation.
	 \item Suppose that $G$ is finite.  Prove that every equivalence class of $\sim$ has size $|H|$.  Conclude that $|H|$ divides $|G|$. 
	\end{enumerate}

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	\item For each pair of groups, demonstrate an isomorphism between them or prove that they are not isomorphic.
	\begin{enumerate}
	  \item $(\ZZ/4\ZZ, +)$ and $(\{1,-1,i,-i\},\cdot)$.
	  \item $\mathfrak{S}_3$ and $(\ZZ/6\ZZ, +)$.
	  \item $G$ and $H$ defined in Problem \ref{GH}.
	  \item $(\ZZ/7\ZZ \setminus \{\overline{0}\},\cdot)$ and $(\ZZ/6\ZZ, +)$.
	 \end{enumerate}

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	\item Let $G$ and $H$ be groups with $e$ the identity element of $H$.  For group homomorphism $f:G \to H$, the {\em kernel} of $f$, denoted $\ker(f)$, is defined as
	 \[ \ker(f) = \{ g\in G \mid f(g) = e \}. \]
	Prove that $\ker(f)$ is a subgroup of $G$.

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\end{enumerate}
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