## Problem 227

Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.
Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.

Then show that $N$ contains all $p$-Sylow subgroups of $G$.

## Problem 226

Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.
Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.
Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.

## Problem 225

Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.

## Problem 224

In the ring
$\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},$ show that $5$ is a prime element but $7$ is not a prime element.

## Problem 223

Consider the ring
$\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}$ and its ideal
$P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.$ Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.

## Problem 222

Suppose that $n\times n$ matrices $A$ and $B$ are similar.

Then show that the nullity of $A$ is equal to the nullity of $B$.
In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of $B$.

## Problem 221

Let $p$ be a prime number. Let
$G=\{z\in \C \mid z^{p^n}=1\}$ be the group of $p$-power roots of $1$ in $\C$.

Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.
Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ itself.

## Problem 220

Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.

## Problem 219

Use Lagrange’s Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat’s Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.

## Problem 218

For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
$A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.$

(a) Find the determinant of the matrix $A$.

(b) Show that $A$ is an orthogonal matrix.

(c) Find the eigenvalues of $A$.

## Problem 217

Let $A, B, C$ are $2\times 2$ diagonalizable matrices.

The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.

From this information, determine the rank of the matrices $A, B,$ and $C$.

Graphs of characteristic polynomials

## Problem 216

Let
$A=\begin{bmatrix} 1 & 3 & 3 \\ -3 &-5 &-3 \\ 3 & 3 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 4 & 3 \\ -4 &-6 &-3 \\ 3 & 3 & 1 \end{bmatrix}.$ For this problem, you may use the fact that both matrices have the same characteristic polynomial:
$p_A(\lambda)=p_B(\lambda)=-(\lambda-1)(\lambda+2)^2.$

(a) Find all eigenvectors of $A$.

(b) Find all eigenvectors of $B$.

(c) Which matrix $A$ or $B$ is diagonalizable?

(d) Diagonalize the matrix stated in (c), i.e., find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$ or $B=PDP^{-1}$.

(Stanford University Linear Algebra Final Exam Problem)

## Problem 215

Show that fields $\Q(\sqrt{2}+\sqrt{3})$ and $\Q(\sqrt{2}, \sqrt{3})$ are equal.

## Problem 214

Find the inverse matrix of the matrix
$A=\begin{bmatrix} \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt] \frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt] -\frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{bmatrix}.$

## Problem 213

Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.

## Problem 212

Let $G$ be a group. Suppose that the order of nonidentity element of $G$ is $2$.
Then show that $G$ is an abelian group.

## Problem 211

In this post, we explain how to diagonalize a matrix if it is diagonalizable.

As an example, we solve the following problem.

Diagonalize the matrix
$A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(Update 10/15/2017. A new example problem was added.)

## Problem 210

Let $A$ be an $n\times n$ matrix with real number entries.

Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.

## Problem 209

Let $G$ be a group. We fix an element $x$ of $G$ and define a map
$\Psi_x: G\to G$ by mapping $g\in G$ to $xgx^{-1} \in G$.
Then prove the followings.
(a) The map $\Psi_x$ is a group homomorphism.

(b) The map $\Psi_x=\id$ if and only if $x\in Z(G)$, where $Z(G)$ is the center of the group $G$.

(c) The map $\Psi_y=\id$ for all $y\in G$ if and only if $G$ is an abelian group.

## Problem 208

Let $G, G’$ be groups and let $f:G \to G’$ be a group homomorphism.
Put $N=\ker(f)$. Then show that we have
$f^{-1}(f(H))=HN.$