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\begin{center}
{\large MATH 108 Fall 2019 \ - \ {\bf Problem Set 10}}
% Put your name and section here.
{\large due December 6}
\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 The {\em dihedreal group of the square}, $D_4$, is the group of the symmetries of a square. Let $e \in D_4$ be the identity element. Let $r \in D_4$ denote a $90^\circ$ counter-clockwise rotation of the square. Let $s \in D_4$ denote a reflection of the square across a vertical line through the center. List the eight elements of $D_4$ in terms of $r$ and $s$ and find the order of each element. (You can physically model $D_4$ by rotating and flipping a square of paper.)
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\item Let $G$ be a group (represented multiplicatively) and let $f:G \to G$ be the function defined by $f(x) = x^{-1}$. Prove that $f$ is a group homomorphism if and only if $G$ is abelian.
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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 $(\ZZ/5\ZZ \setminus \{\overline{0}\},\cdot)$ and $(\ZZ/4\ZZ, +)$.
\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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\item Let $G$ be a finite 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 Prove that every equivalence class of $\sim$ has cardinality $|H|$.
\item Prove that $|H|$ divides $|G|$.
\end{enumerate}
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\end{enumerate}
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