## $p$-Group Acting on a Finite Set and the Number of Fixed Points

## Problem 359

Let $P$ be a $p$-group acting on a finite set $X$.

Let

\[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \]

The prove that

\[|X^P|\equiv |X| \pmod{p}.\]

Let

\[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \]

The prove that

\[|X^P|\equiv |X| \pmod{p}.\]

Let $\alpha= \sqrt[3]{2}e^{2\pi i/3}$. Prove that $x_1^2+\cdots +x_k^2=-1$ has no solutions with all $x_i\in \Q(\alpha)$ and $k\geq 1$.

Add to solve later Let $A$ be an $n\times n$ matrix. Assume that every vector $\mathbf{x}$ in $\R^n$ is an eigenvector for some eigenvalue of $A$.

Prove that there exists $\lambda\in \R$ such that $A=\lambda I$, where $I$ is the $n\times n$ identity matrix.

**(a)** Let $S=\{\mathbf{v}_1, \mathbf{v}_2\}$ be the set of the following vectors in $\R^4$.

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

1 \\

0 \\

1 \\

0

\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}

0 \\

1 \\

1 \\

0

\end{bmatrix}.\]
Find an orthogonal basis of the subspace $\Span(S)$ of $\R^4$.

**(b)** Let $T:\R^2 \to \R^3$ be a linear transformation such that

\[T(\mathbf{e}_1)=\mathbf{u}_1 \text{ and } T(\mathbf{e}_2)=\mathbf{u}_2,\]
where $\{\mathbf{e}_1, \mathbf{e}_2\}$ is the standard unit vectors of $\R^2$ and

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

5 \\

1 \\

2

\end{bmatrix} \text{ and } \mathbf{u}_2=\begin{bmatrix}

8 \\

2 \\

6

\end{bmatrix}.\]
Then find

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

3 \\

-2

\end{bmatrix} \,\right).\]

Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$.

Prove the **Cauchy-Schwarz inequality**:

\[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\]

Let $G$ be a group. Let $a$ and $b$ be elements of $G$.

If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.

Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying

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

1 \\

2

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

3 \\

4 \\

5

\end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}

0 \\

1

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

0 \\

0 \\

1

\end{bmatrix}.\]
Find a general formula for

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

x_1 \\

x_2

\end{bmatrix} \,\right).\]

(*The Ohio State University, Linear Algebra Math 2568 Exam Problem*)

A **hyperplane ** in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors

\[\begin{bmatrix}

x_1 \\

x_2 \\

\vdots \\

x_n

\end{bmatrix}\in \R^n\]
satisfying the linear equation of the form

\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\]
where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.

Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation

\[a_1x_1+a_2x_2+\cdots+a_nx_n=0,\]
where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.

(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

Add to solve laterLet $R$ be a commutative ring with unity.

Then show that every maximal ideal of $R$ is a prime ideal.

Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.

Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of $S$.

\[[v_1]_B=\begin{bmatrix}

1 \\

0 \\

0 \\

0

\end{bmatrix}, [v_2]_B=\begin{bmatrix}

0 \\

1 \\

0 \\

0

\end{bmatrix}, [v_3]_B=\begin{bmatrix}

1 \\

1 \\

0 \\

0

\end{bmatrix}.\]

Let $V$ be the vector space of all $2\times 2$ real matrices.

Let $S=\{A_1, A_2, A_3, A_4\}$, where

\[A_1=\begin{bmatrix}

1 & 2\\

-1& 3

\end{bmatrix}, A_2=\begin{bmatrix}

0 & -1\\

1& 4

\end{bmatrix}, A_3=\begin{bmatrix}

-1 & 0\\

1& -10

\end{bmatrix}, A_4=\begin{bmatrix}

3 & 7\\

-2& 6

\end{bmatrix}.\]
Then find a basis for the span $\Span(S)$.

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

Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as

\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.

Let $C$ be the companion matrix of the polynomial $p(x)$ given by

\[C=\begin{bmatrix}

0 & 0 & \dots & 0 &-a_0 \\

1 & 0 & \dots & 0 & -a_1 \\

0 & 1 & \dots & 0 & -a_2 \\

\vdots & & \ddots & & \vdots \\

0 & 0 & \dots & 1 & -a_{n-1}

\end{bmatrix}=

[\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],\]
where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by

\[\mathbf{a}=\begin{bmatrix}

a_0 \\

a_1 \\

\vdots \\

a_{n-1}

\end{bmatrix}.\]

Then prove that the following two statements are equivalent.

- There exists a vector $\mathbf{v}\in \C^n$ such that

\[\mathbf{v}, A\mathbf{v}, A^2\mathbf{v}, \dots, A^{n-1}\mathbf{v}\] form a basis of $\C^n$. - There exists an invertible matrix $S$ such that $S^{-1}AS=C$.

(Namely, $A$ is similar to the companion matrix of its characteristic polynomial.)

Let $V$ be a vector space over a scalar field $K$.

Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ be the set of vectors in $V$, where $n \geq 2$.

Then prove that the set $S$ is linearly dependent if and only if at least one of the vectors in $S$ can be written as a linear combination of remaining vectors in $S$.

Add to solve later Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.

Suppose that $G$ does not have a normal subgroup of order $3$.

Then determine all group homomorphisms from $G$ to $K$.

Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.

Let $I$ be the subset of $R$ defined by

\[I:=\{ f(x) \in R \mid f(1)=0\}.\]

Then prove that $I$ is an ideal of the ring $R$.

Moreover, show that $I$ is maximal and determine $R/I$.

Let $a, b$ be relatively prime integers and let $p$ be a prime number.

Suppose that we have

\[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\]
for some positive integer $n$.

Then prove that $2^{n+1}$ divides $p-1$.

Add to solve later Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.

Let $\Aut(N)$ be the group of automorphisms of $G$.

Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.

Then prove that $N$ is contained in the center of $G$.

Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.

Prove that we have an isomorphism of groups:

\[G \cong \ker(f)\times \Z.\]

Let $H$ and $K$ be normal subgroups of a group $G$.

Suppose that $H < K$ and the quotient group $G/H$ is abelian.

Then prove that $G/K$ is also an abelian group.

Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$.

Then prove that the quotient group $G/N$ is also an abelian group.