## 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 $3$.

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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 $3$.

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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 group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$.

**(a)** Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$.

**(b)** Prove that a group cannot be written as the union of two proper subgroups.

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 $N=\{b^m \mid b\in G\}$.

Suppose that $p$ is a prime number greater than $3$.

Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$.

**(a)** Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$.

**(b)** Determine the index $[G : S]$.

**(c)** Assume that $-1\notin S$. Then prove that for each $a\in G$ we have either $a\in S$ or $-a\in S$.

Let $G$ 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 $\phi$.

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 later 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 $x, y$ 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 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 laterLet $G$ be a nilpotent group and let $H$ be a proper subgroup of $G$.

Then prove that $H \subsetneq N_G(H)$, where $N_G(H)$ is the normalizer of $H$ in $G$.

Add to solve later Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is,

\[H=\{ a\in G \mid \text{the order of $a$ is finite}\}.\]

Prove that $H$ is a subgroup of $G$.

Add to solve laterLet $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.

Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.

Add to solve later Let $G$ be an abelian group.

Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.

Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.

Also determine whether the statement is true if $G$ is a non-abelian group.

Add to solve laterProve that every finite group having more than two elements has a nontrivial automorphism.

(*Michigan State University, Abstract Algebra Qualifying Exam*)

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