Algebra Qual 2023-09

Problem 1

Let $G$ be a group, and let $Aut (G)$ denote the group of automorphisms of $G$ . There is a homomorphism $γ : G \to Aut (G)$ that takes $s \in G$ to the automorphism $γ_{s}$ defined by $γ_{s} (t) = s t s^{- 1}$ .

Prove rigorously, possibly with induction, that is $γ_{s} (t) = t^{b}$ , then $γ_{s^{n}} (t) = t^{b^{n}}$ .
Suppose $s \in G$ has order 5, and $s t s^{- 1} = t^{2}$ . Find the order of $t$ . Justify your answer.

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Let $G$ be a group, and let $\operatorname{Aut}(G)$ denote the group of automorphisms of $G$. There is a homomorphism $\gamma:G\to \operatorname{Aut}(G)$ that takes $s\in G$ to the automorphism $\gamma_s$ defined by $\gamma_s(t)=sts^{-1}$.
	\begin{enumerate}[label=\alph*)]
		\item Prove rigorously, possibly with induction, that is $\gamma_s(t)=t^b$, then $\gamma_{s^n}(t)=t^{b^n}$.
		\item Suppose $s\in G$ has order 5, and $sts^{-1}=t^2$. Find the order of $t$. Justify your answer.
	\end{enumerate}

Problem 2

Suppose $G$ is a nonempty finite set that has an associative pairing $G \times G \to G$ , written $(x, y) \mapsto x \cdot y$ . Suppose this pairing satisfies left and right cancellation: $x \cdot y = x \cdot y^{'}$ implies $y = y^{'}$ , and $x \cdot y = x^{'} \cdot y$ implies $x = x^{'}$ . Prove there exists an element $e$ of $G$ such that for all $x \in G$ , $e \cdot x = x \cdot e = x$ . Justify your reasoning as carefully as possible.

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Suppose $G$ is a nonempty finite set that has an associative pairing $G\times G\to G$, written $(x,y)\mapsto x\cdot y$. Suppose this pairing satisfies left and right cancellation: $x\cdot y = x\cdot y'$ implies $y=y'$, and $x\cdot y = x'\cdot y$ implies $x=x'$. Prove there exists an element $e$ of $G$ such that for all $x\in G$, $e\cdot x = x\cdot e = x$. Justify your reasoning as carefully as possible.

Problem 3

Let $R_{1}, \dots, R_{k}$ be commutative rings, and set $R = R_{1} \times \dots \times R_{k}$ .

Let $I_{j} \subset R_{j}$ be ideals, and put $I = I_{1} \times \dots \times I_{k}$ . Use the First Isomorphism Theorem to prove that $R / I ≃ R_{1} / I_{1} \times \dots \times R_{k} / I_{k}$ .
Prove the prime ideals of $R$ have the form $R_{1} \times \dots \times R_{j - 1} \times P_{j} \times R_{j + 1} \times \dots \times R_{k}$ where $P_{j} \subset R_{j}$ is a prime ideal for $1 \leq j \leq k$ . (Omit the proof that this is an ideal.)

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Let $R_1,\ldots, R_k$ be commutative rings, and set $R=R_1\times \cdots \times R_k$.
	\begin{enumerate}[label=\alph*)]
		\item Let $I_j\subset R_j$ be ideals, and put $I=I_1\times \cdots \times I_k$. Use the First Isomorphism Theorem to prove that $R/I\simeq R_1/I_1\times \cdots \times R_k/I_k$.
		\item Prove the prime ideals of $R$ have the form $R_1\times \cdots \times R_{j-1}\times P_j\times R_{j+1}\times \cdots \times R_k$ where $P_j\subset R_j$ is a prime ideal for $1\leq j\leq k$. (Omit the proof that this is an ideal.)
	\end{enumerate}

Problem 4

Let $i$ be the imaginary number, let $Z [i] = {a + b i ∣ a, b \in Z}$ , a principal ideal domain, and let $Z_{2}$ be the finite ring of integers modulo 2.

Define a ring homomorphism from $Z [i] \to Z_{2}$ . You must prove it is a ring homomorphism.
Find, with proof, a generator for the kernel of your ring homomorphism.

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Let $i$ be the imaginary number, let ${\bf Z}[i]=\{a+bi\,\mid \, a,b\in {\bf Z}\}$, a principal ideal domain, and let ${\bf Z}_2$ be the finite ring of integers modulo 2.
	\begin{enumerate}[label=\alph*)]
		\item Define a ring homomorphism from ${\bf Z}[i]\to {\bf Z}_2$. You must prove it is a ring homomorphism.
		\item Find, with proof, a generator for the kernel of your ring homomorphism.
	\end{enumerate}

Problem 5

Suppose $T : R^{n} \to R^{n}$ is a linear transformation with distinct eigenvalues $λ_{1}, λ_{2}, \dots, λ_{m}$ , and let $v_{1}, v_{2}, \dots, v_{m}$ be corresponding eigenvectors. Prove $v_{1}, v_{2}, \dots, v_{m}$ are linearly independent.

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Suppose $T:{\bf R}^n\to {\bf R}^n$ is a linear transformation with distinct eigenvalues $\lambda_1, \lambda_2,\ldots, \lambda_m$, and let ${\bf v}_1,{\bf v}_2,\ldots, {\bf v}_m$ be corresponding eigenvectors. Prove ${\bf v}_1,{\bf v}_2,\ldots, {\bf v}_m$ are linearly independent.