## The Center of a p-Group is Not Trivial

## Problem 10

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.

Read solution

of the day

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.

Read solution

Let $A$ 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 $T(\mathbf{0}_n)=\mathbf{0}_m$.

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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