Algebra Qual 2022-06

Problem 1

Let $N$ be a finite normal subgroup of $G$ . Prove there is a normal subgroup $M$ of $G$ such that $[G : M]$ is finite and $n m = m n$ for all $n \in N$ and $m \in M$ .

Hint: You may use the fact that the centralizer $C (h) := {g \in G ∣ g h g^{- 1} = h}$ is a subgroup of $G$ .)

View

L A T E X

code

Let $N$ be a finite normal subgroup of $G$. Prove there is a normal subgroup $M$ of $G$ such that $[G:M]$ is finite and $nm=mn$ for all $n\in N$ and $m\in M$.

\medskip
\noindent ({\itshape Hint:} You may use the fact that the centralizer $C(h):=\{g\in G\mid ghg^{-1}=h\}$ is a subgroup of $G$.)

Problem 2

Let $S_{7}$ denote the symmetric group.

Give an example of two non-conjugate elements of $S_{7}$ that have the same order.
If $g \in S_{7}$ has maximal order, what is the order of $g$ ?
Does the element $g$ that you found in part (2) lie in $A_{7}$ ? Fully justify your answer.
Determine whether the set ${h \in S_{7} ∣ | h | = | g |}$ is a single conjugacy class in $S_{7}$ , where $g$ is the element you found in part (2).

View

L A T E X

code

Let $S_7$ denote the symmetric group.
\begin{enumerate}[label=\alph*)]
	\item Give an example of two non-conjugate elements of $S_7$ that have the same order.
	\item If $g\in S_7$ has maximal order, what is the order of $g$?
	\item Does the element $g$ that you found in part (b) lie in $A_7$? Fully justify your answer.
	\item Determine whether the set $\{h\in S_7\mid |h|=|g|\}$ is a single conjugacy class in $S_7$, where $g$ is the element you found in part (b).
\end{enumerate}

Problem 3

Let $R$ be a commutative ring with $1$ . Use theorems in ring theory to prove:

$⟨ x ⟩$ is a prime ideal in $R [x]$ if and only if $R$ is an integral domain.
$⟨ x ⟩$ is a maximal ideal in $R [x]$ if and only if $R$ is a field.

View

L A T E X

code

Let $R$ be a commutative ring with $1$. Use theorems in ring theory to prove:
\begin{enumerate}[label=\alph*)]
	\item $\langle x\rangle$ is a prime ideal in $R[x]$ if and only if $R$ is an integral domain.
	\item $\langle x\rangle$ is a maximal ideal in $R[x]$ if and only if $R$ is a field.
\end{enumerate}

Problem 4

Let $R$ be a commutative ring with $1$ , and $σ : R \to R$ be a ring automorphism.

Show that $F = {r \in R ∣ σ (r) = r}$ is a subring of $R$ (with $1$ ).
Show that if $σ^{2}$ is the identity map on $R$ , then each element of $R$ is the root of a monic polynomial of degree 2 in $F [x]$ , where $F$ is as in (a).

View

L A T E X

code

Let $R$ be a commutative ring with $1$, and $\sigma:R\to R$ be a ring automorphism.
\begin{enumerate}[label=\alph*)]
	\item Show that $F=\{r\in R\mid \sigma(r)=r\}$ is a subring of $R$ (with $1$).
	\item Show that if $\sigma^2$ is the identity map on $R$, then each element of $R$ is the root of a monic polynomial of degree 2 in $F[x]$, where $F$ is as in (a).
\end{enumerate}

Problem 5

Let $A = [\begin{matrix} 2 & - 1 & - 1 \\ 1 & 0 & - 1 \\ 1 & - 1 & 0 \end{matrix}]$ .

Compute the characteristic polynomial $p_{A} (x)$ of $A$ . It has integer roots.
For each eigenvalue $λ$ of $A$ , find a basis for the eigenspace $E_{λ}$ .
Determine if $A$ is diagonalizable. If so, give matrices $P$ and $B$ such that $P^{- 1} A P = B$ and $B$ is diagonal. If no, explain carefully why $A$ is not diagonalizable.

View

L A T E X

code

Let $A=\begin{bmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Compute the characteristic polynomial $p_A(x)$ of $A$. It has integer roots.
	\item For each eigenvalue $\lambda$ of $A$, find a basis for the eigenspace $E_{\lambda}$.
	\item Determine if $A$ is diagonalizable. If so, give matrices $P$ and $B$ such that $P^{-1}AP=B$ and $B$ is diagonal. If no, explain carefully why $A$ is not diagonalizable.
\end{enumerate}