Algebra Qual 2022-09

Problem 1

Let $G$ be a group and $N$ a normal subgroup of $G$ . Let $a N$ denote the left coset defined by $a \in G$ , and consider the binary operation

$G / N \times G / N \to G / N$

given by $(a N, b N) \mapsto a b N$ .

Show the operation is well defined.
Show the operation is well defined only if the subgroup $N$ is normal.

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L A T E X

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Let $G$ be a group and $N$ a normal subgroup of $G$. Let $aN$ denote the left coset defined by $a\in G$, and consider the binary operation
\[
	G/N\times G/N\to G/N
\]
given by $(aN, bN)\mapsto abN$.
\begin{enumerate}[label=\alph*)]
	\item Show the operation is well defined.
	\item Show the operation is well defined only if the subgroup $N$ is normal.
\end{enumerate}

Problem 2

Let $C$ be a (possibly infinite) cyclic group, and let $Aut (C)$ and $Inn (C)$ be the groups of automorphisms and inner automorphisms, respectively. (Recall an automorphism $γ$ is inner if it is given by conjugation: $γ (b) = a b a^{- 1}$ for some $a \in C$ .)

Describe $Aut (C)$ and $Inn (C)$ in familiar terms, as groups you would study in a first algebra course. Prove your result. (Hint: Where do generators go?)
Write $Aut (Z_{12})$ down explicitly, giving its generic name and computing the order of every element. Show all work.

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Let $C$ be a (possibly infinite) cyclic group, and let $\operatorname{Aut}(C)$ and $\operatorname{Inn}(C)$ be the groups of automorphisms and inner automorphisms, respectively. (Recall an automorphism $\gamma$ is {\bfseries inner} if it is given by conjugation: $\gamma(b)=aba^{-1}$ for some $a\in C$.)
\begin{enumerate}[label=\alph*)]
	\item Describe $\operatorname{Aut}(C)$ and $\operatorname{Inn(C)}$ in familiar terms, as groups you would study in a first algebra course. Prove your result. ({\itshape Hint:} Where do generators go?)
	\item Write $\operatorname{Aut}({\bf Z}_{12})$ down explicitly, giving its generic name and computing the order of every element. Show all work.
\end{enumerate}

Problem 3

Let $R$ be a commutative ring with $1$ . The characteristic $char (R)$ of $R$ is the unique integer $n \geq 0$ such that $⟨ n ⟩ \subset Z$ is the kernel of the homomorphism $θ : Z \to R$ given by

$θ (m) = {\begin{cases} \underset{m}{\underset{⏟}{1_{R} + \dots + 1_{R}}}, & if m \geq 0 \\ \underset{| m |}{\underset{⏟}{- 1_{R} + \dots + - 1_{R}}}, & if m < 0 \end{cases}$

Prove that if $f : R \to S$ is a monomorphism of commutative rings with $1$ , then $char (R) = char (S)$ .
Prove by given an example that $char (R)$ is not always preserved by ring homomorphisms.

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L A T E X

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Let $R$ be a commutative ring with $1$. The {\bfseries characteristic} $\operatorname{char}(R)$ of $R$ is the unique integer $n\geq 0$ such that $\langle n\rangle \subset {\bf Z}$ is the kernel of the homomorphism $\theta:{\bf Z}\to R$ given by
\[
	\theta(m)=\begin{cases} \underbrace{1_R+\cdots +1_R}_{m}, & \text{ if }m\geq 0 \\ \underbrace{-1_R+\cdots+-1_R}_{|m|}, & \text{ if }m<0\end{cases}
\]
\begin{enumerate}[label=\alph*)]
	\item Prove that if $f:R\to S$ is a monomorphism of commutative rings with $1$, then $\operatorname{char}(R)=\operatorname{char}(S)$.
	\item Prove by given an example that $\operatorname{char}(R)$ is not always preserved by ring homomorphisms.
\end{enumerate}

Problem 4

Let $W \subset R^{5}$ be the space spanned by the vectors

${[\begin{matrix} 1 \\ - 2 \\ 0 \\ 2 \\ 1 \end{matrix}], [\begin{matrix} - 2 \\ 4 \\ - 1 \\ 1 \\ 2 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 2 \\ - 2 \\ 1 \end{matrix}]} .$

Compute the dimension of $W$ .
Let $W^{⊥} = {v \in R^{5} ∣ v \cdot w = 0 for all w \in W}$ . Determine the dimension of $W^{⊥}$ , and explain how this following immediately from (a) using a theorem.
Find a basis for $W^{⊥}$ .

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L A T E X

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Let$W\subset {\bf R}^5$ be the space spanned by the vectors
\[
	\left\{\begin{bmatrix} 1 \\ -2 \\ 0 \\ 2 \\ 1\end{bmatrix},\begin{bmatrix} -2 \\ 4 \\ -1 \\ 1 \\ 2\end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 2 \\ -2 \\1\end{bmatrix}\right\}.
\]
\begin{enumerate}[label=\alph*)]
	\item Compute the dimension of $W$.
	\item Let $W^{\perp}=\{{\bf v}\in {\bf R}^5\,\mid\, {\bf v}\cdot {\bf w}=0\text{ for all }w\in W\}$. Determine the dimension of $W^{\perp}$, and explain how this following immediately from (a) using a theorem.
	\item Find a basis for $W^{\perp}$.
\end{enumerate}

Problem 5

Let $T : R^{3} \to R^{3}$ be the orthogonal projection to a $1$ -dimensional linear subspace $L \subset R^{3}$ .

List the eigenvalues of $T$ .
Write the characteristic polynomial $p_{T} (x)$ for $T$ .
Is $T$ diagonalizable? Briefly justify your answer.

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Let $T:{\bf R}^3\to {\bf R}^3$ be the orthogonal projection to a $1$-dimensional linear subspace $L\subset {\bf R}^3$.
\begin{enumerate}[label=\alph*)]
	\item List the eigenvalues of $T$.
	\item Write the characteristic polynomial $p_T(x)$ for $T$.
	\item Is $T$ diagonalizable? Briefly justify your answer.
\end{enumerate}