Algebra Qual 2023-01

Problem 1

Let $G$ be a finite group and $n > 1$ an integer such that $(a b)^{n} = a^{n} b^{n}$ for all $a, b \in G$ . Let

$G_{n} = {c \in G ∣ c^{n} = e} and G^{n} = {c^{n} ∣ c \in G}$

You may take for granted that these are subgroups. Prove that both $G_{n}$ and $G^{n}$ are normal in $G$ , and $| G^{n} | = [G : G_{n}]$ .

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Let $G$ be a finite group and $n>1$ an integer such that $(ab)^n=a^n b^n$ for all $a,b\in G$. Let
 \[
	 G_n=\{c\in G\mid c^n=e\}\qquad\text{and}\qquad G^n=\{c^n\mid c\in G\}
\]
You may take for granted that these are subgroups. Prove that both $G_n$ and $G^n$ are normal in $G$, and $|G^n|=[G:G_n]$.

Problem 2

Show that every finite group with more than two elements has a nontrivial automorphism.

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Show that every finite group with more than two elements has a nontrivial automorphism.

Problem 3

Let $R$ be a commutative ring with identity. Suppose that for every $a \in R$ there is an integer $n \geq 2$ such that $a^{n} = a$ . Show that every prime ideal of $R$ is maximal.

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Let $R$ be a commutative ring with identity.  Suppose that for every $a\in R$ there is an integer $n\geq 2$ such that $a^n=a$. Show that every prime ideal of $R$ is maximal.

Problem 4

Let $M_{n} (R)$ be the vector space of all $n \times n$ matrices with real entries. We say that $A, B \in M_{n} (R)$ commute if $A B = B A$ .

Fix $A \in M_{n} (R)$ . Prove that the set of all matrices in $M_{n} (R)$ that commute with $A$ is a subspace of $M_{n} (R)$ .
Let $A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] \in M_{2} (R)$ and let $W \subseteq M_{2} (R)$ be the subspace of all matrices of $M_{2} (R)$ that commute with $A$ . Find a basis of $W$ .

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Let $\operatorname{M}_n({\bf R})$ be the vector space of all $n \times n$ matrices with real entries. We say that $A, B \in \operatorname{M}_n({\bf R})$ commute if $AB = BA$.
\begin{enumerate}[label=\alph*)]
	\item Fix $A \in \operatorname{M}_n({\bf R})$. Prove that the set of all matrices in $\operatorname{M}_n({\bf R})$ that commute with $A$ is a subspace of $\operatorname{M}_n({\bf R})$.
	\item Let $A=\begin{bmatrix} 1 & 1 \\ 1 & 1  \end{bmatrix}\in \operatorname{M}_2({\bf R})$ and let $W\subseteq \operatorname{M}_2({\bf R})$ be the subspace of all matrices of $\operatorname{M}_2({\bf R})$ that commute with $A$. Find a basis of $W$. 
\end{enumerate}

Problem 5

Let $S : V \to V$ and $T : V \to V$ be linear transformations that commute, i.e. $S \circ T = T \circ S$ . Let $v \in V$ be an eigenvector of $S$ such that $T (v) \neq 0$ . Prove that $T (v)$ is also an eigenvector of $S$ .

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Let $S : V \to V$ and $T : V \to V$ be linear transformations that commute, i.e. $S \circ T = T \circ S$. Let $v \in V$ be an eigenvector of $S$ such that $T(v) \ne 0$. Prove that $T(v)$ is also an eigenvector of $S$.