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\begin{document}
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\begin{center}\textbf{Math 31 Take Home Midterm}\\Due July 25, 2018\end{center}

\textbf{Instructions:} This is an open notes, open book exam. You may use your notes or homework assignments from class, as well as Pinter 2nd edition. You may not use the internet or any other textbooks, nor may you confer with classmates. If you use any unauthorized aid, this is an honor code violation. This exam is due at the beginning of class on July 25.

\begin{enumerate}

\item Let $G$ be a group of permutations on a set $X$, and let $a\in X$. Then $$\text{Stab}(a)=\{\alpha\in G : \alpha(a)=a\}$$ is called the \textbf{stabilizer} of $a$ in $G$. Prove that $\text{Stab}(a)$ is a subgroup of $G$.

\item Prove that if a subgroup has index 2, then it is a normal subgroup.

\item Recall that the centralizer of an element $a\in G$ is defined by $$C(a)=\{g\in G : ga=ag\}.$$ Find the centralizer of $(123)$ in $S_3$.

\item If $a,g\in G$ where $G$ is a group, prove that $C(a)\cong C(gag^{-1})$.

\item Prove that $\left<\{2^m3^n : m,n\in\Z\},\cdot\right>$ is not cyclic.

\item Let $G$ be a group of order $25$. Prove that either $G$ is cyclic, or $g^5=e$ for all $g\in G$.

\item Let $H$ be a subgroup of $G$, and let $a\in G$ be a fixed element.
\begin{enumerate}
\item Prove that $aHa^{-1}$ is a subgroup of $G$ of the same order as $H$.
\item Let $|H|=n$ and let $H$ be the unique subgroup of $G$ of order $n$. Prove that $H\unlhd G$.
\end{enumerate}
\end{enumerate}
\end{document}