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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due March 31 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

%%------------------------------------------------------------------------

\begin{exercise} % HW 10, Problem 1
Let $(G,*)$ be a group.
\begin{enumerate}[label=(\alph*)]
    \item Re-write the definition of a group in terms of quantifiers.
    \item Prove that the identity element is unique.
    \item Prove that each element has a unique inverse. 
\end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 10, Problem 2
Let $H_1$ and $H_2$ be subgroups of a group $G$. 
\begin{enumerate}[label=(\alph*)]
    \item Prove that $H_1\cap H_2$ is a subgroup of $G$.
    \item Does your argument extend to $\cap_{\alpha\in I}H_\alpha$, where  $\{H_\alpha:\alpha\in I\}$ is an arbitrary (not necessarily finite) collection of subgroups? If yes, prove it. If not, give a counterexample.
\end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 10, Problem 3
Let $\phi\colon G\to H$ be a homomorphism, and $K\leq G$ a subgroup. Prove that the \emph{image} of $K$, denoted $\phi(K)=\{\phi(g)\mid g\in G\}$, is a subgroup of $H$.
%\begin{enumerate}[label=(\alph*)]
%\end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 10, Problem 4
Let $\phi\colon G\to H$ be a homomorphism, and denote the identity elements as $e_G$ and $e_H$, respectively. Define the \emph{kernel} of $\phi$ to be the set of elements of $G$ that are sent to the identity in $H$:
\[
\ker(\phi)=\big\{k\in G\mid \phi(k)=e_H\big\}.
\]
Prove that $\ker(\phi)$ is a subgroup of $G$.
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 10, Problem 5
    Let $G$ be a group. The \emph{center} of $G$, denoted $Z(G)$, is the set of all elements that commute with everything. That is,
    \[
    Z(G)=\big\{z\in G\mid zg=gz\,\;\text{for all $g\in G$}\big\}.
    \]
    \begin{enumerate}[label=(\alph*)]
        \item Find the center of each of the following groups: $\Z$, $D_4$, and $Q_8$.
        \item Prove that $Z(G)$ is a subgroup of $G$.
        \item For each $x\in G$, the \emph{centralizer} of $x$, is the set
        $C_G(x)=\{g\in G\mid gx=xg\}$. Prove that
        \[
        Z(G)=\bigcap_{x\in G}C_G(x).
        \]
    \end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 10, Problem 6
Verify that the function
\[
\phi\colon\C\longto M_2(\R),\qquad \phi(a+bi)=\begin{bmatrix}a&b\\-b&a\end{bmatrix}
\]
is a ring homomorphism. That is, show that for any $z=a+bi$ and $w=c+di$, both $\phi(z+w)=\phi(z)+\phi(w)$ and $\phi(zw)=\phi(z)\phi(w)$ hold.
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\end{document}

