## Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8

## Problem 568

Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.

Add to solve laterProve that every group of order $24$ has a normal subgroup of order $4$ or $8$.

Add to solve later Prove the following identity for any positive integer $n$.

\[\begin{bmatrix}

\cos \theta & -\sin \theta\\

\sin \theta& \cos \theta

\end{bmatrix}^n=\begin{bmatrix}

\cos n\theta & -\sin n\theta\\

\sin n\theta& \cos n\theta

\end{bmatrix}.\]

Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.

Add to solve later Let $I$ be the $2\times 2$ identity matrix.

Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.

Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$.

**(a)** Prove that $A+B$ is skew-symmetric.

**(b)** Prove that $cA$ is skew-symmetric for any scalar $c$.

**(c)** Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is skew-symmetric.

**(d)** Suppose that $A$ is real skew-symmetric. Prove that $iA$ is an Hermitian matrix.

**(e)** Prove that if $AB=-BA$, then $AB$ is a skew-symmetric matrix.

**(f)** Let $\mathbf{v}$ be an $n$-dimensional column vecotor. Prove that $\mathbf{v}^{\trans}A\mathbf{v}=0$.

**(g)** Suppose that $A$ is a real skew-symmetric matrix and $A^2\mathbf{v}=\mathbf{0}$ for some vector $\mathbf{v}\in \R^n$. Then prove that $A\mathbf{v}=\mathbf{0}$.

Let

\[\mathbf{v}_1=\begin{bmatrix}

1 \\

2 \\

0

\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}

1 \\

a \\

5

\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}

0 \\

4 \\

b

\end{bmatrix}\]
be vectors in $\R^3$.

Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.

Add to solve later An $n\times n$ matrix $A$ is called **nonsingular** if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.

Using the definition of a nonsingular matrix, prove the following statements.

**(a)** If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular.

**(b)** Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then:

- The matrix $B$ is nonsingular.
- The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.)

Let $A$ be a singular $n\times n$ matrix.

Let

\[\mathbf{e}_1=\begin{bmatrix}

1 \\

0 \\

\vdots \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1 \\

\vdots \\

0

\end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}

0 \\

0 \\

\vdots \\

1

\end{bmatrix}\]
be unit vectors in $\R^n$.

Prove that at least one of the following matrix equations

\[A\mathbf{x}=\mathbf{e}_i\]
for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.

Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.

Then the product $A\mathbf{b}$ is an $n$-dimensional vector.

Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.

Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.

Add to solve laterFor each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$.

**(a)** $A=\begin{bmatrix}

4 & 2\\

2& 1

\end{bmatrix}$.

**(b)** $A=\begin{bmatrix}

2 & 1\\

1& 3

\end{bmatrix}$.

Let $A$ be an $n\times n$ nonsingular matrix.

Prove that the transpose matrix $A^{\trans}$ is also nonsingular.

Add to solve later Let $N$ be a normal subgroup of a group $G$.

Suppose that $G/N$ is an infinite cyclic group.

Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.

Add to solve later Let $\mathbf{v}$ be a nonzero vector in $\R^n$.

Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$.

Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by

\[A=I-a\mathbf{v}\mathbf{v}^{\trans},\]
where $I$ is the $n\times n$ identity matrix.

Prove that $A$ is a symmetric matrix and $AA=I$.

Conclude that the inverse matrix is $A^{-1}=A$.

Let $U$ and $V$ be vector spaces over a scalar field $\F$.

Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$.

**(a)** Prove that $T:U\to V$ is a linear transformation.

(Hence, $T$ is called the **zero transformation**.)

**(b)** Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$.

Let $x, y$ be generators of a group $G$ with relation

\begin{align*}

xy^2=y^3x,\tag{1}\\

yx^2=x^3y.\tag{2}

\end{align*}

Prove that $G$ is the trivial group.

Add to solve later Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula

\[T\left(\, \begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix} \,\right)=\begin{bmatrix}

x_1+3x_2-2x_3 \\

2x_1+3x_2 \\

x_2-x_3

\end{bmatrix}.\]

Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$.

Add to solve laterFor each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.

**(a)** $A=\begin{bmatrix}

1 & 3 & -2 \\

2 &3 &0 \\

0 & 1 & -1

\end{bmatrix}$

**(b)** $A=\begin{bmatrix}

1 & 0 & 2 \\

-1 &-3 &2 \\

3 & 6 & -2

\end{bmatrix}$.

Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$.

Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$.

Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$.

Prove that

\[\cos ^2\theta_1+\cdots+\cos^2 \theta_n=1.\]

Consider the $2\times 2$ matrix

\[A=\begin{bmatrix}

\cos \theta & -\sin \theta\\

\sin \theta& \cos \theta \end{bmatrix},\]
where $\theta$ is a real number $0\leq \theta < 2\pi$.

**(a)** Find the characteristic polynomial of the matrix $A$.

**(b)** Find the eigenvalues of the matrix $A$.

**(c)** Determine the eigenvectors corresponding to each of the eigenvalues of $A$.

By calculating the **Wronskian**, determine whether the set of exponential functions

\[\{e^x, e^{2x}, e^{3x}\}\]
is linearly independent on the interval $[-1, 1]$.