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\begin{center}
{\large MATH 150A Winter 2020 \ - \ {\bf Problem Set 4}}
% Put your name and section here.
{\large due February 7}
\end{center}
\begin{enumerate}
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\item Let $G$ be a group with identity element $e$.
\begin{enumerate}
\item Prove that $G/G$ is trivial.
\item Prove that $G/\{e\}$ is isomorphic to $G$.
\end{enumerate}
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\item Let $G$ be the additive group of $\RR^2$ and $H$ the subgroup
\[ H = \{(x,x) \mid x \in \RR\}. \]
\begin{enumerate}
\item Characterize all left cosets of $H$.
\item Prove that $G/H$ is isomorphic to the additive group of $\RR$.
\item Find a homomorphism $f:\RR^2 \to \RR$ with $\ker f = H$.
\end{enumerate}
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\item (2.12.4) Let $G = \CC^\times$ and $H = \{\pm 1, \pm i\}$, the subgroup of fourth roots of unity.
\begin{enumerate}
\item Characterize all left cosets of $H$.
\item Prove or disprove that $G/H$ isomorphic to $G$.
\end{enumerate}
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\item (2.12.1) Show that if a subgroup $H$ of a group $G$ is not normal, then there are left cosets $aH$ and $bH$ whose product is not a coset of $H$.
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\item Let $K$ be a normal subgroup of $G$ and $q:G \to G/K$ be the quotient map. Let $f:G \to H$ be a homomorphism with $K \subseteq \ker(f)$. Prove that $f$ factors through $q$, meaning that there exists a homomorphism $\varphi:G/K \to H$ such that $f = \varphi \circ q$.
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\item (2.11.1) Let $x$ be an element of group $G$ with order $r$, and let $y$ be an elements of group $H$ with order $s$. Find the order of $(x,y)$ in the product group $G \times H$.
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\item (2.11.9) Let $H$ and $K$ be subroups of group $G$. Prove that the product set $HK$ is a subgroup of $G$ if and only if $HK = KH$.
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\item Let $G$ and $H$ be groups.
Prove that $G \times \{1\}$ is a normal subgroup of $G\times H$.
Prove that $(G\times H)/(G \times \{1\})$ is isomorphic to $H$.
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\end{enumerate}
\end{document}