Algebra Qual 2014-03

Problem 1

Let $G$ be a group and suppose $Aut (G)$ is trivial.

Show that $G$ is abelian.
Show that for any abelian group $H$ , the inversion map $ϕ (h) = h^{- 1}$ is an automorphism.
Use parts (1) and (2) above to show that $g^{2}$ is the identity element for every $g \in G$ .

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Let $G$ be a group and suppose $\operatorname{Aut}(G)$ is trivial.
\begin{enumerate}[label=(\alph*)]
	\item Show that $G$ is abelian.
	\item Show that for any abelian group $H$, the {\bfseries inversion map} $\phi(h)=h^{-1}$ is an automorphism.
	\item Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.
\end{enumerate}

Problem 2

Let $G$ be the group of upper-triangular real matrices $[\begin{matrix} a & b \\ 0 & d \end{matrix}]$ with $a, d \neq 0$ , under matrix multiplication. Let $S$ be the subset of $G$ defined by $d = 1$ . Show that $S$ is normal and that $G / S ≅ R^{\times}$ , the multiplicative group of nonzero real numbers.

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Let $G$ be the group of upper-triangular real matrices $\begin{bmatrix} a & b \\ 0 & d\end{bmatrix}$ with $a,d\neq 0$, under matrix multiplication. Let $S$ be the subset of $G$ defined by $d=1$. Show that $S$ is normal and that $G/S\cong {\bf R}^{\times}$, the multiplicative group of nonzero real numbers.

Problem 3

Let $A$ be a commutative ring with unit. We call $A$ Boolean if $a^{2} = a$ for every $a \in A$ . Prove that in a Boolean ring $A$ each of the following holds:

$2 a = 0$ for every $a \in A$ .
If $I$ is a prime ideal then $A / I$ is a field with two elements (and in particular $I$ is maximal).
If $I = (a, b)$ is the ideal generated by $a$ and $b$ then $I$ can be generated by the single element $a + b + a b$ . Conclude that every finitely generated ideal is principal.

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Let $A$ be a commutative ring with unit. We call $A$ {\bfseries Boolean} if $a^2=a$ for every $a\in A$. Prove that in a Boolean ring $A$ each of the following holds:
\begin{enumerate}[label=(\alph*)]
	\item $2a=0$ for every $a\in A$.
	\item If $I$ is a prime ideal then $A/I$ is a field with two elements (and in particular $I$ is maximal).
	\item If $I=(a,b)$ is the ideal generated by $a$ and $b$ then $I$ can be generated by the single element $a+b+ab$. Conclude that every finitely generated ideal is principal.
\end{enumerate}

Problem 4

Let $a, b \in R$ and $T : R^{3} \to R^{3}$ be the linear transformation which is reflection across the plane $z = a x + b y$ .

Find the eigenvalues of $T$ and for each find a basis for the corresponding eigenspace.
Is $T$ diagonalizable? Justify.
What is the characteristic polynomial of $T$ ?
What is the minimal polynomial of $T$ ?

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Let $a,b\in {\bf R}$ and $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation which is reflection across the plane $z=ax+by$.
\begin{enumerate}[label=(\alph*)]
	\item Find the eigenvalues of $T$ and for each find a basis for the corresponding eigenspace.
	\item Is $T$ diagonalizable? Justify.
	\item What is the characteristic polynomial of $T$?
	\item What is the minimal polynomial of $T$?
\end{enumerate}

Problem 5

Let $ϕ : V \to W$ be a surjective linear transformation of finite-dimensional linear spaces. Show that there is a $U \subseteq V$ such that $V = (\ker (ϕ)) \oplus U$ and $ϕ ∣_{U} : U \to W$ is an isomorphism. (Note that $V$ is not assumed to be an inner-product space; also note that $\ker (ϕ)$ is sometimes referred to as the null space of $ϕ$ ; finally, $ϕ ∣_{U}$ denotes the restriction of $ϕ$ to $U$ .)

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Let $\phi: V\to W$ be a surjective linear transformation of finite-dimensional linear spaces. Show that there is a $U\subseteq V$ such that $V=(\ker(\phi))\oplus U$ and $\phi\mid_U:U\to W$ is an isomorphism. (Note that $V$ is not assumed to be an inner-product space; also note that $\ker(\phi)$ is sometimes referred to as the {\bfseries null space} of $\phi$; finally, $\phi\mid_U$ denotes the restriction of $\phi$ to $U$.)