## If the Quotient by the Center is Cyclic, then the Group is Abelian

## Problem 18

Let $Z(G)$ be the center of a group $G$.

Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian.

Let $Z(G)$ be the center of a group $G$.

Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian.

Let $p$ be a prime number. Suppose that the order of each element of a finite group $G$ is a power of $p$. Then prove that $G$ is a $p$-group. Namely, the order of $G$ is a power of $p$.

Add to solve laterLet $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?

**(a)** At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ for $i=1,2,3,4$.

**(b)** At $0$ each of the polynomials has the value $1$. Namely $p_i(0)=1$ for $i=1,2,3,4$.

(*University of California, Berkeley*)

Here is a very short true or false problem.

Select either *True* or *False*. Then click “*Finish quiz*” button.

You will be able to see an explanation of the solution by clicking “*View questions*” button.

Let $A$ and $B$ be $n\times n$ matrices.

Suppose that these matrices have a common eigenvector $\mathbf{x}$.

Show that $\det(AB-BA)=0$.

Read solution

Let $A$ be an $n \times n$ real matrix. Prove the followings.

**(a)** The matrix $AA^{\trans}$ is a symmetric matrix.

**(b) **The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.

**(c)** The matrix $AA^{\trans}$ is non-negative definite.

(An $n\times n$ matrix $B$ is called *non-negative definite* if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)

**(d)** All the eigenvalues of $AA^{\trans}$ is non-negative.

An $n\times n$ matrix $A$ is called **nilpotent** if $A^k=O$, where $O$ is the $n\times n$ zero matrix.

Prove the followings.

**(a)** The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.

**(b)** The matrix $A$ is nilpotent if and only if $A^n=O$.

Read solution

Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$.

(Such a group is called a $p$*-group*.)

Show that the center $Z(G)$ of the group $G$ is not trivial.

Add to solve laterLet $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.

Show that

**(1) ** $$\det(A)=\prod_{i=1}^n \lambda_i$$

**(2)** $$\tr(A)=\sum_{i=1}^n \lambda_i$$

Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$.

Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues.

Read solution

Let $A= \begin{bmatrix}

1 & 2\\

2& 1

\end{bmatrix}$.

Compute $A^n$ for any $n \in \N$.

Let $A=\begin{bmatrix}

a & 0\\

0& b

\end{bmatrix}$.

Show that

**(1)** $A^n=\begin{bmatrix}

a^n & 0\\

0& b^n

\end{bmatrix}$ for any $n \in \N$.

**(2) **Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix.

Show that $B^n=S^{-1}A^n S$ for any $n \in \N$

Define the functions $f_{a,b}(x)=ax+b$, where $a, b \in \R$ and $a>0$.

Show that $G:=\{ f_{a,b} \mid a, b \in \R, a>0\}$ is a group . The group operation is function composition.

Add to solve laterLet $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.

Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.

Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.

(*The Ohio State University Linear Algebra Exam*)

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Let $G$ and $G’$ be a group and let $\phi:G \to G’$ be a group homomorphism.

Show that $\phi$ induces an injective homomorphism from $G/\ker{\phi} \to G’$.

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Let $H$ be a normal subgroup of a group $G$.

Then show that $N:=[H, G]$ is a subgroup of $H$ and $N \triangleleft G$.

Here $[H, G]$ is a subgroup of $G$ generated by commutators $[h,k]:=hkh^{-1}k^{-1}$.

In particular, the commutator subgroup $[G, G]$ is a normal subgroup of $G$

Add to solve laterShow that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.

Add to solve laterA square matrix $A$ is called **idempotent** if $A^2=A$.

Show that a square invertible idempotent matrix is the identity matrix.

Add to solve later