Algebra Qual 2020-06

Let $G$ be a group of order $2p$, where $p$ is an odd prime. Prove $G$ contains a nontrivial, proper normal subgroup.

Suppose $G$ is a nontrivial finite group and $H,K\mathrel{\unlhd}G$ are normal subgroups with $\gcd(|H|,|K|)=1$.
\begin{enumerate}[label=\alph*)]
	\item Define a nontrivial group homomorphism $\phi:G\to G/H\times G/K$
	\item Prove $G$ is isomorphic to a subgroup of $G/H\times G/K$.
	\item Suppose $\gcd(m,n)=1$. Prove ${\bf Z}_{mn}\cong {\bf Z}_m\times {\bf Z}_n$.
\end{enumerate}

Suppose $R$ is a ring such that $r^2=r$ for every element $r\in R$.
\begin{enumerate}[label=\alph*)]
	\item Prove $r=-r$ for every element $r\in R$.
	\item Show $R$ must be commutative. {\itshape Hint:} Consider $(a+b)^2$.
\end{enumerate}

Let ${\bf Z}\left[\sqrt{2}\right]=\left\{a+b\sqrt{2}\,|\, a,b\in{ \bf Z}\right\}$, and let $R\subset \operatorname{M}_2({\bf Z})$ be the ring of all $2\times 2$ matrices of the form $\begin{bmatrix} a & b \\ 2b & a\end{bmatrix}$. Prove ${\bf Z}\left[\sqrt{2}\right]$ is isomorphic to $R$.

\medskip
\noindent {\itshape Hint:} Start by showing the map $a+b\sqrt{2}\mapsto \begin{bmatrix} a & b \\ 2b & a\end{bmatrix}$ is a ring homomorphism.

A real $n\times n$ matrix $A$ is called {\bfseries skew-symmetric} if $A^{\top}=-A$. Let $V_n$ be the set of all skew-symmetric matrices in $\operatorname{M}_n({\bf R})$. Recall that $\operatorname{M}_n({\bf R})$ is an $n^2$-dimensional ${\bf R}$-vector space with standard basis $\left\{e_{ij}\,|\, 1\leq i,j\leq n\right\}$, where $e_{ij}$ is the $n\times n$ matrix with a 1 in the $(i,j)$-position and zeros everywhere else.
\begin{enumerate}[label=\alph*)]
	\item Show $V_n$ is a subspace of $\operatorname{M}_n({\bf R})$.
	\item Find an ordered basis $\mathcal{B}$ for the space $V_3$ of all skew-symmetric $3\times 3$ matrices.
\end{enumerate}