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 $| G^{n} | = [G : G_{n}]$ .

Problem 2

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.

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 $A B = B A$ .
(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 $M_{n} (R)$ .
(b) 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 $W$ .

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 $S$ .