# Category: Group Theory

## The Group of Rational Numbers is Not Finitely Generated

## Problem 461

**(a)** Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated.

**(b)** Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.

## 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.

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

## The Order of a Conjugacy Class Divides the Order of the Group

## Problem 455

Let $G$ be a finite group.

The **centralizer** of an element $a$ of $G$ is defined to be

\[C_G(a)=\{g\in G \mid ga=ag\}.\]

A **conjugacy class** is a set of the form

\[\Cl(a)=\{bab^{-1} \mid b\in G\}\]
for some $a\in G$.

**(a)**Prove that the centralizer of an element of $a$ in $G$ is a subgroup of the group $G$.

**(b)** Prove that the order (the number of elements) of every conjugacy class in $G$ divides the order of the group $G$.

## The Product of a Subgroup and a Normal Subgroup is a Subgroup

## Problem 448

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

The **product** of $H$ and $N$ is defined to be the subset

\[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\]
Prove that the product $H\cdot N$ is a subgroup of $G$.

## Inverse Map of a Bijective Homomorphism is a Group Homomorphism

## Problem 445

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

Suppose that $f:G\to H$ is bijective.

Then there exists a map $\psi:H\to G$ such that

\[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\]
Then prove that $\psi:H \to G$ is also a group homomorphism.

## Group Homomorphism Sends the Inverse Element to the Inverse Element

## Problem 444

Let $G, G’$ be groups. Let $\phi:G\to G’$ be a group homomorphism.

Then prove that for any element $g\in G$, we have

\[\phi(g^{-1})=\phi(g)^{-1}.\]

## Injective Group Homomorphism that does not have Inverse Homomorphism

## Problem 443

Let $A=B=\Z$ be the additive group of integers.

Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$.

**(a)** Prove that $\phi$ is a group homomorphism.

**(b)** Prove that $\phi$ is injective.

**(c)** Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.

## Fundamental Theorem of Finitely Generated Abelian Groups and its application

## Problem 420

In this post, we study the **Fundamental Theorem of Finitely Generated Abelian Groups**, and as an application we solve the following problem.

**Problem**.

Let $G$ be a finite abelian group of order $n$.

If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic to the cyclic group $Z_n=\Zmod{n}$ of order $n$.

## Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$

## Problem 402

Let $G$ be a group. Suppose that we have

\[(ab)^3=a^3b^3\]
for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$.

Then prove that $G$ is an abelian group.

Add to solve later## Prove a Group is Abelian if $(ab)^2=a^2b^2$

## Problem 401

Let $G$ be a group. Suppose that

\[(ab)^2=a^2b^2\]
for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.

## $p$-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

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

## Order of Product of Two Elements in a Group

## Problem 354

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.

## Group Homomorphisms From Group of Order 21 to Group of Order 49

## Problem 346

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 $K$.

## Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$.

## Problem 344

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## Abelian Normal subgroup, Quotient Group, and Automorphism Group

## Problem 343

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 $G$.

## Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups

## Problem 342

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:

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

## If Quotient $G/H$ is Abelian Group and $H < K \triangleleft G$, then $G/K$ is Abelian

## Problem 341

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.

## Quotient Group of Abelian Group is Abelian

## Problem 340

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.