# Tagged: generator

## If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial

## Problem 554

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## Prove that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse Elements

## Problem 529

Let $\F_3=\Zmod{3}$ be the finite field of order $3$.

Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$.

**(a)** Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have?

**(b)** Let $ax+b+I$ be a nonzero element of the field $\F_3[x]/(x^2+1)$, where $a, b \in \F_3$. Find the inverse of $ax+b+I$.

**(c)** Recall that the multiplicative group of nonzero elements of a field is a cyclic group.

Confirm that the element $x$ is not a generator of $E^{\times}$, where $E=\F_3[x]/(x^2+1)$ but $x+1$ is a generator.

Add to solve later## Every Finitely Generated Subgroup of Additive Group $\Q$ of Rational Numbers is Cyclic

## Problem 460

Let $\Q=(\Q, +)$ be the additive group of rational numbers.

**(a)** Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.

**(b)** Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.

## Prove that a Group of Order 217 is Cyclic and Find the Number of Generators

## Problem 458

Let $G$ be a finite group of order $217$.

**(a)** Prove that $G$ is a cyclic group.

**(b)** Determine the number of generators of the group $G$.

## A Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator

## Problem 434

Let $R$ be a ring with $1$.

A nonzero $R$-module $M$ is called **irreducible** if $0$ and $M$ are the only submodules of $M$.

(It is also called a **simple** module.)

**(a)** Prove that a nonzero $R$-module $M$ is irreducible if and only if $M$ is a cyclic module with any nonzero element as its generator.

**(b)** Determine all the irreducible $\Z$-modules.

## Generators of the Augmentation Ideal in a Group Ring

## Problem 302

Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by

\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\]
where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring homomorphism, called the **augmentation map** and the kernel of $\epsilon$ is called the **augmentation ideal**.

**(a)** Prove that the augmentation ideal in the group ring $RG$ is generated by $\{g-e \mid g\in G\}$.

**(b)** Prove that if $G=\langle g\rangle$ is a finite cyclic group generated by $g$, then the augmentation ideal is generated by $g-e$.

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## A Simple Abelian Group if and only if the Order is a Prime Number

## Problem 290

Let $G$ be a group. (Do not assume that $G$ is a finite group.)

Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.

## Commutator Subgroup and Abelian Quotient Group

## Problem 147

Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$.

Let $N$ be a subgroup of $G$.

Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.

## Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup

## Problem 129

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

For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$.

Let $[H,K]$ be a subgroup of $G$ generated by all such commutators.

Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup $[H, K]$ is normal in $G$.

Add to solve later## If a Subgroup $H$ is in the Center of a Group $G$ and $G/H$ is Nilpotent, then $G$ is Nilpotent

## Problem 128

Let $G$ be a nilpotent group and let $H$ be a subgroup such that $H$ is a subgroup in the center $Z(G)$ of $G$.

Suppose that the quotient $G/H$ is nilpotent.

Then show that $G$ is also nilpotent.

Add to solve later## Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$

## Problem 53

Let $D_8$ be the dihedral group of order $8$.

Using the generators and relations, we have

\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\]

**(a)** Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$.

Prove that the centralizer $C_{D_8}(A)=A$.

**(b)** Show that the normalizer $N_{D_8}(A)=D_8$.

**(c) **Show that the center $Z(D_8)=\langle r^2 \rangle=\{1,r^2\}$, the subgroup generated by $r^2$.

## Dihedral Group and Rotation of the Plane

## Problem 52

Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by

\[D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.\]
Put $\theta=2 \pi/n$.

**(a)**Prove that the matrix $\begin{bmatrix}

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

\sin \theta& \cos \theta

\end{bmatrix}$ is the matrix representation of the linear transformation $T$ which rotates the $x$-$y$ plane about the origin in a counterclockwise direction by $\theta$ radians.

**(b)**Let $\GL_2(\R)$ be the group of all $2 \times 2$ invertible matrices with real entries. Show that the map $\rho: D_{2n} \to \GL_2(\R)$ defined on the generators by

\[ \rho(r)=\begin{bmatrix}

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

\sin \theta& \cos \theta

\end{bmatrix} \text{ and }

\rho(s)=\begin{bmatrix}

0 & 1\\

1& 0

\end{bmatrix}\] extends to a homomorphism of $D_{2n}$ into $\GL_2(\R)$.

**(c)**Determine whether the homomorphism $\rho$ in part (b) is injective and/or surjective. Add to solve later

## A Condition that a Commutator Group is a Normal Subgroup

## Problem 3

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 later