## 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 finite group and let $N$ be a normal subgroup of $G$.

Suppose that the order $n$ of $N$ is relatively prime to the index $|G:N|=m$.

**(a)** Prove that $N=\{a\in G \mid a^n=e\}$.

**(b)** Prove that $N=\{b^m \mid b\in G\}$.

Is it possible that each element of an infinite group has a finite order?

If so, give an example. Otherwise, prove the non-existence of such a group.

Let $G$ be a finite group of order $2n$.

Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup.

Namely, suppose that $G=S\sqcup H$, where $S$ is the set of all elements of order in $G$, and $H$ is a subgroup of $G$. The cardinalities of $S$ and $H$ are both $n$.

Then prove that $H$ is an abelian normal subgroup of odd order.

Add to solve laterProve that every group of order $24$ has a normal subgroup of order $4$ or $8$.

Add to solve laterLet $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.

Add to solve laterLet $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$.

Prove that the product $HK$ can never be a subgroup of the group $G$.

Add to solve later Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying

\[|A|+|B| > |G|.\]
Here $|X|$ denotes the cardinality (the number of elements) of the set $X$.

Then prove that $G=AB$, where

\[AB=\{ab \mid a\in A, b\in B\}.\]

Let $G$ be a finite group and let $S$ be a non-empty set.

Suppose that $G$ acts on $S$ freely and transitively.

Prove that $|G|=|S|$. That is, the number of elements in $G$ and $S$ are the same.

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 $a\in G$.

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

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 that if $G$ is a finite group of even order, then the number of elements of $G$ of order $2$ is odd.

Add to solve laterLet $R$ 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 finite group. Let $a, b$ be elements of $G$.

Prove that the order of $ab$ is equal to the order of $ba$.

(Of course do not assume that $G$ is an abelian group.)

Read solution

Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.

Add to solve laterLet $p, q$ be prime numbers such that $p>q$.

If a group $G$ has order $pq$, then show the followings.

**(a)** The group $G$ has a normal Sylow $p$-subgroup.

**(b)** The group $G$ is solvable.

Use Lagrange’s Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat’s Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.

Add to solve laterLet $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.

Read solution

Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that

\[b^m=a.\]

Read solution

Let $p$ be a prime number.

Let $G$ be a non-abelian $p$-group.

Show that the index of the center of $G$ is divisible by $p^2$.

Read solution