## Problem 118

Let $G$ be a finite group of order $18$.

Show that the group $G$ is solvable.

## Problem 117

Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$.
Let $H$ be a subgroup of $G$ containing the normalizer $N_G(P)$ of $P$ in $G$.

Then show that $N_G(H)=H$.

## Problem 116

Let $G$ and $G’$ be groups and let $f:G \to G’$ be a group homomorphism.
If $H’$ is a normal subgroup of the group $G’$, then show that $H=f^{-1}(H’)$ is a normal subgroup of the group $G$.

## Problem 115

Express the vector $\mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 \end{bmatrix}$ as a linear combination of the vectors
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2= \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{v}_3= \begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}.$

(The Ohio State University, Linear Algebra Exam)

## Problem 114

Let
$A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.$ Compute $A^{2017}\mathbf{u}$.

(The Ohio State University, Linear Algebra Exam)

## Problem 113

Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.
The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation
$(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),$ where $a_i \in A, b_i \in B$ for $i=1, 2$.

Let $f: A \to A’$ and $g:B \to B’$ be group isomorphisms. Define $\phi’: B’\to \Aut(A’)$ by sending $b’ \in B’$ to $f\circ \phi(g^{-1}(b’))\circ f^{-1}$.

$\require{AMScd} \begin{CD} B @>{\phi}>> \Aut(A)\\ @A{g^{-1}}AA @VV{\sigma_f}V \\ B’ @>{\phi’}>> \Aut(A’) \end{CD}$ Here $\sigma_f:\Aut(A) \to \Aut(A’)$ is defined by $\alpha \in \Aut(A) \mapsto f\alpha f^{-1}\in \Aut(A’)$.
Then show that
$A \rtimes_{\phi} B \cong A’ \rtimes_{\phi’} B’.$

## Problem 112

Let $G$ be a simple group and let $X$ be a finite set.
Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$.
Then show that $G$ is a finite group and the order of $G$ divides $|X|!$.

## Problem 111

Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.

(a) The product $AB$ is symmetric if and only if $AB=BA$.

(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.

## Problem 110

Let $p \in \Z$ be a prime number.

Then describe the elements of the Galois group of the polynomial $x^p-2$.

## Problem 109

Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$.

For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.

## Problem 108

Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.
Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, the $n$-dimensional vector space over $\F_p$. Therefore, $G_n$ acts on $\F_p^n$.

Let $e_n \in \F_p^n$ be the vector $(1,0, \dots,0)$.
(The so-called first standard basis vector in $\F_p^n$.)

Find the size of the $G_n$-orbit of $e_n$, and show that $\Stab_{G_n}(e_n)$ has order $|G_{n-1}|\cdot p^{n-1}$.

Conclude by induction that
$|G_n|=p^{n^2}\prod_{i=1}^{n} \left(1-\frac{1}{p^i} \right).$

## Problem 107

For what value(s) of $a$ does the system have nontrivial solutions?
\begin{align*}
&x_1+2x_2+x_3=0\\
&-x_1-x_2+x_3=0\\
& 3x_1+4x_2+ax_3=0.
\end{align*}

## Problem 106

Let $G$ be a finite group of odd order. Assume that $x \in G$ is not the identity element.

Show that $x$ is not conjugate to $x^{-1}$.

## Problem 105

Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$.

Then prove that any subgroup of index $p$ is a normal subgroup of $G$.

## Problem 104

Test your understanding of basic properties of matrix operations.

There are 10 True or False Quiz Problems.

These 10 problems are very common and essential.
So make sure to understand these and don’t lose a point if any of these is your exam problems.
(These are actual exam problems at the Ohio State University.)

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.

## Problem 103

Find the rank of the following real matrix.
$\begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},$ where $a$ is a real number.

(Kyoto University, Linear Algebra Exam)

## Problem 102

Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.

(a) $\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.

(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.

(c) A homogeneous system of $3$ equations in $4$ unknowns.

(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
$\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$ (The Ohio State University, Linear Algebra Exam)

## Problem 101

For which choice(s) of the constant $k$ is the following matrix invertible?
$A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.$
(Johns Hopkins University, Linear Algebra Exam)

## Problem 100

Determine whether a group $G$ of the following order is simple or not.

(a) $|G|=100$.
(b) $|G|=200$.

## Problem 99

Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.