## Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57

## Problem 628

Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.

Then determine the number of elements in $G$ of order $3$.

Read solution

Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.

Then determine the number of elements in $G$ of order $3$.

Read solution

Let $G$ be a group. Suppose that the number of elements in $G$ of order $5$ is $28$.

Determine the number of distinct subgroups of $G$ of order $5$.

Add to solve laterLet $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$.

Prove that the number of elements in $S$ is odd.

Add to solve laterLet $m$ and $n$ be positive integers such that $m \mid n$.

**(a)** Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined.

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

**(c)** Prove that $\phi$ is surjective.

**(d)** Determine the group structure of the kernel of $\phi$.

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

Suppose that $G/N$ is an infinite cyclic group.

Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.

Add to solve later 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 Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.

Then prove that $R$ is a commutative ring.

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.

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 $G$.

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.

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 $n$.

If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module.

Can this action be extended to make $M$ into a $\Q$-module?

Add to solve laterLet $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.

Prove that we have an isomorphism of groups:

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

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 $g-e$.

Read solution

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.

Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.

Add to solve laterShow that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.

Add to solve laterLet $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.

Add to solve later