\documentclass{article}
\usepackage{amsmath, amssymb,verbatim, amsthm, graphicx}

%exercise environment
\newcounter{exercise}
\setcounter{exercise}{0}
\newenvironment{exercise}{\addtocounter{exercise}{1} \noindent{\bf{Exercise \theexercise.}}}{\vspace{.5cm}}

\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\mat}[4]{\left( \begin{array}{cc} #1 & #2 \\ #3 & #4  \end{array} \right)}
\newcommand{\smat}[4]{\left( \begin{smallmatrix} #1 & #2 \\ #3 & #4  \end{smallmatrix} \right)}

\textwidth 6.5in
\textheight 9in
\oddsidemargin 0in
\evensidemargin 0in
\headsep -0.5in

\setlength{\parskip}{0.25cm}

\pagenumbering{gobble}

\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
\end{center}

\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 12.3  \\ Spring 2019 \hfill Due May 1}}}

\large

\vspace{1cm}


\begin{exercise}
Let $G$ be a group acting on a set $S$.  Prove that for any $s \in S$, 
\[
\operatorname{Stab}(s) = \{ x \in G \, | \, xs = s \}
\]
is a subgroup of $G$.
\end{exercise}

\begin{exercise}  
Let $G$ be a group acting on a set $S$.  Prove that if $s, t \in S$ and $t= xs$, then
\[
\operatorname{Stab}(t) = x\operatorname{Stab}(s)x^{-1}.
\]
\end{exercise}

\begin{exercise}
Recall that an action of a group $G$ on a set $S$ gives rise to a homomorphism $\pi : G \to \operatorname{Perm}(S)$, where $\pi_x(s) = xs$ for all $x \in G$ and $s \in S$.  Prove that
\[
\ker \pi = \bigcap_{s \in S} \operatorname{Stab}(s).
\]
\end{exercise}


\end{document}
