## Group Homomorphism, Preimage, and Product of Groups

## 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

\[f^{-1}(f(H))=HN.\]

Let $G, G’$ be groups and let $f:G \to G’$ be a group homomorphism.

Put $N=\ker(f)$. Then show that we have

\[f^{-1}(f(H))=HN.\]

Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$.

Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.

Determine all eigenvalues and their algebraic multiplicities of the matrix

\[A=\begin{bmatrix}

1 & a & 1 \\

a &1 &a \\

1 & a & 1

\end{bmatrix},\]
where $a$ is a real number.

Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order $m$ and $n$, respectively.

If $m$ and $n$ are relatively prime, then show that the order of the element $ab$ is $mn$.

Is there a (not necessarily commutative) ring $R$ with $1$ such that the equation

\[x+x=1 \]
has more than one solutions $x\in R$?

Let $R$ be a commutative ring. Let $S$ be a subset of $R$ and let $I$ be an ideal of $I$.

We define the subset

\[(I:S):=\{ a \in R \mid aS\subset I\}.\]
Prove that $(I:S)$ is an ideal of $R$. This ideal is called the * ideal quotient*, or

Show that eigenvalues of a Hermitian matrix $A$ are real numbers.

(*The Ohio State University Linear Algebra Exam Problem*)

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Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$.

Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.

Add to solve laterLet

\[ A=\begin{bmatrix}

5 & 2 & -1 \\

2 &2 &2 \\

-1 & 2 & 5

\end{bmatrix}.\]

Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.

Your score of this problem is equal to that dimension times five.

(*The Ohio State University Linear Algebra Practice Problem*)

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Let $R$ be the ring of all continuous functions on the interval $[0,1]$.

Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$.

Show that the set $I$ is an ideal of $R$ but is not a prime ideal.

Add to solve laterLet $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.

Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.

Add to solve laterLet $R$ be a ring with unit $1\neq 0$.

Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.

(Do not assume that the ring $R$ is commutative.)

Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$. Then for any elements $h \in H$ and $k\in K$, show that $hk=kh$.

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Let $G$ be a group and let $A$ be an abelian subgroup of $G$ with $A \triangleleft G$.

(That is, $A$ is a normal subgroup of $G$.)

If $B$ is any subgroup of $G$, then show that

\[A \cap B \triangleleft AB.\]

Find the value(s) of $h$ for which the following set of vectors

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

1 \\

0 \\

0

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

h \\

1 \\

-h

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

1 \\

2h \\

3h+1

\end{bmatrix}\right\}\]
is linearly independent.

(*Boston College, Linear Algebra Midterm Exam Sample Problem*)

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Let $A$ be a $3 \times 3$ matrix.

Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have

\[A\mathbf{x}=\begin{bmatrix}

1 \\

0 \\

1

\end{bmatrix}, A\mathbf{y}=\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}, A\mathbf{z}=\begin{bmatrix}

1 \\

1 \\

1

\end{bmatrix}.\]

Then find the value of the determinant of the matrix $A$.

Add to solve laterLet

\[A=\begin{bmatrix}

1 & -1\\

2& 3

\end{bmatrix}.\]

Find the eigenvalues and the eigenvectors of the matrix

\[B=A^4-3A^3+3A^2-2A+8E.\]

(*Nagoya University Linear Algebra Exam Problem*)

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Prove that the matrix

\[A=\begin{bmatrix}

1 & 1.00001 & 1 \\

1.00001 &1 &1.00001 \\

1 & 1.00001 & 1

\end{bmatrix}\]
has one positive eigenvalue and one negative eigenvalue.

(*University of California, Berkeley Qualifying Exam Problem*)

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Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces

\[E_2=\Span\left \{\quad \begin{bmatrix}

1 \\

1 \\

1 \\

1

\end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix}

1 \\

2 \\

1 \\

1

\end{bmatrix},\quad \begin{bmatrix}

1 \\

1 \\

1 \\

2

\end{bmatrix} \quad\right\}.\]

Calculate $C^4 \mathbf{u}$ for $\mathbf{u}=\begin{bmatrix}

6 \\

8 \\

6 \\

9

\end{bmatrix}$ if possible. Explain why if it is not possible!

(*The Ohio State University Linear Algebra Exam Problem*)

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