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 $(a N, b N) \mapsto a b N$ .
(a) Show the operation is well defined.
(b) Show the operation is well defined only if the subgroup $N$ is normal.

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 $a \in C$ .)
(a) 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?)
(b) Write $Aut (Z_{12})$ down explicitly, giving its generic name and computing the order of every element. Show all work.

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}

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

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}]} .

(a) Compute the dimension of $W$ .
(b) 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.
(c) Find a basis for $W^{⊥}$ .

Problem 5

Let $T : R^{3} \to R^{3}$ be the orthogonal projection to a $1$ -dimensional linear subspace $L \subset R^{3}$ .
(a) List the eigenvalues of $T$ .
(b) Write the characteristic polynomial $p_{T} (x)$ for $T$ .
(c) Is $T$ diagonalizable? Briefly justify your answer.