Algebra Qual 2020-06

Problem 1

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

Problem 2

Suppose $G$ is a nontrivial finite group and $H,K⊴G$ are normal subgroups with $gcd(|H|,|K|)=1$.
(a) Define a nontrivial group homomorphism $\varphi :G\to G/H\times G/K$.
(b) Prove $G$ is isomorphic to a subgroup of $G/H\times G/K$.
(c) Suppose $gcd(m,n)=1$. Prove ${\mathbf{Z}}_{mn}\cong {\mathbf{Z}}_m\times {\mathbf{Z}}_n$.

Problem 3

Suppose $R$ is a ring such that $r^2=r$ for every element $r\in R$.
(a) Prove $r=-r$ for every element $r\in R$.
(b) Show $R$ must be commutative. Hint: Consider $(a+b{)}^2$.

Problem 4

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

Hint: Start by showing the map $a+b\sqrt{2}\mapsto \left[\begin{array}{cc}a& b\\ 2b& a\end{array}\right]$ is a ring homomorphism.

Problem 5

A real $n\times n$ matrix $A$ is called skew-symmetric if $A^{\mathrm{\top }}=-A$. Let $V_n$ be the set of all skew-symmetric matrices in ${\mathrm{M}}_n(\mathbf{R})$. Recall that ${\mathrm{M}}_n(\mathbf{R})$ is an $n^2$-dimensional $\mathbf{R}$-vector space with standard basis $\{e_{ij}|1\le i,j\le n\}$, where $e_{ij}$ is the $n\times n$ matrix with a 1 in the $(i,j)$-position and zeros everywhere else.
(a) Show $V_n$ is a subspace of ${\mathrm{M}}_n(\mathbf{R})$.
(b) Find an ordered basis $\mathcal{B}$ for the space $V_3$ of all skew-symmetric $3\times 3$ matrices.