Chapter Questions
Find all generators of $Z_{6}, Z_{8}$, and $Z_{20}$.
Suppose that $\langle a\rangle,\langle b\rangle$, and $\langle c\rangle$ are cyclic groups of orders 6,8, and 20, respectively. Find all generators of $\langle a\rangle,\langle b\rangle$, and $\langle c\rangle$.
List the elements of the subgroups $\langle 20\rangle$ and $\langle 10\rangle$ in $Z_{30}$. Let $a$ be a group element of order 30 . List the elements of the subgroups $\left\langle a^{20}\right\rangle$ and $\left\langle a^{10}\right\rangle$.
List the elements of the subgroups $\langle 3\rangle$ and $\langle 15\rangle$ in $Z_{18} .$ Let $a$ be a group element of order 18 . List the elements of the subgroups $\left\langle a^{3}\right\rangle$ and $\left\langle a^{15}\right\rangle$.
List the elements of the subgroups $\langle 3\rangle$ and $\langle 7\rangle$ in $U(20)$.
What do Exercises 3,4, and 5 have in common? Try to make a generalization that includes these three cases.
Find an example of a noncyclic group, all of whose proper subgroups are cyclic.
Let $a$ be an element of a group and let $|a|=15 .$ Compute the orders of the following elements of $G$.a. $a^{3}, a^{6}, a^{9}, a^{12}$b. $a^{5}, a^{10}$c. $a^{2}, a^{4}, a^{8}, a^{14}$
How many subgroups does $Z_{20}$ have? List a generator for each of these subgroups. Suppose that $G=\langle a\rangle$ and $|a|=20$. How many subgroups does $G$ have? List a generator for each of these subgroups.
In $Z_{24}$, list all generators for the subgroup of order 8 . Let $G=\langle a\rangle$ and let $|a|=24$. List all generators for the subgroup of order 8 .
Let $G$ be a group and let $a \in G$. Prove that $\left\langle a^{-1}\right\rangle=\langle a\rangle$.
In $Z$, find all generators of the subgroup $\langle 3\rangle .$ If $a$ has infinite order, find all generators of the subgroup $\left\langle a^{3}\right\rangle$.
Suppose that a cyclic group $G$ has exactly three subgroups: $G$ itself, $\{e\}$, and a subgroup of order 7 . What is $|G|$ ? What can you say if 7 is replaced with $p$ where $p$ is a prime?
Suppose that a cyclic group $G$ has exactly three subgroups: $G$ itself, $\{e\}$, and a subgroup of order 7 . What is $|G| ?$ What can you say if 7 is replaced with $p$ where $p$ is a prime?
Let $G$ be an Abelian group and let $H=\{g \in G|| g \mid$ divides 12$\}$. Prove that $H$ is a subgroup of $G$. Is there anything special about 12 here? Would your proof be valid if 12 were replaced by some other positive integer? State the general result.
Find a collection of distinct subgroups $\left\langle a_{1}\right\rangle,\left\langle a_{2}\right\rangle, \ldots,\left\langle a_{n}\right\rangle$ of $Z_{240}$ with the property that $\left\langle a_{1}\right\rangle \subset\left\langle a_{2}\right\rangle \subset \cdots \subset\left\langle a_{n}\right\rangle$ with $n$ as large aspossible.
Complete the following statement: $|a|=\left|a^{2}\right|$ if and only if $|a| \ldots \ldots$
If a cyclic group has an element of infinite order, how many elements of finite order does it have?
List the cyclic subgroups of $U(30)$.
Suppose that $G$ is an Abelian group of order 35 and every element of $G$ satisfies the equation $x^{35}=e$. Prove that $G$ is cyclic. Does your argument work if 35 is replaced with 33 ?
Let $G$ be a group and let $a$ be an element of $G$.a. If $a^{12}=e$, what can we say about the order of $a$ ?b. If $a^{m}=e$, what can we say about the order of $a$ ?c. Suppose that $|G|=24$ and that $G$ is cyclic. If $a^{8} \neq e$ and $a^{12} \neq e$, show that $\langle a\rangle=G$
Prove that a group of order 3 must be cyclic.
Let $Z$ denote the group of integers under addition. Is every subgroup of $Z$ cyclic? Why? Describe all the subgroups of $Z$. Let $a$ be a group element with infinite order. Describe all subgroups of $\langle a\rangle$.
For any element $a$ in any group $G$, prove that $\langle a\rangle$ is a subgroup of $C(a)$ (the centralizer of $a$ ).
If $d$ is a positive integer, $d \neq 2$, and $d$ divides $n$, show that the number of elements of order $d$ in $D_{n}$ is $\phi(d) .$ How many elements of order 2 does $D_{n}$ have?
Find all generators of $Z$. Let $a$ be a group element that has infinite order. Find all generators of $\langle a\rangle$.
Prove that $C^{*}$, the group of nonzero complex numbers under multiplication, has a cyclic subgroup of order $n$ for every positive integer $n .$
Let $a$ be a group element that has infinite order. Prove that $\left\langle a^{i}\right\rangle=$ $\left\langle a^{j}\right\rangle$ if and only if $i=\pm j$
Suppose $a$ and $b$ belong to a group, $a$ has odd order, and $a b a^{-1}=$ $b^{-1} .$ Show that $b^{2}=e$
Let $G$ be a finite group. Show that there exists a fixed positive integer $n$ such that $a^{n}=e$ for all $a$ in $G$. (Note that $n$ is independent of $a$.)
Determine the subgroup lattice for $Z_{12}$
Determine the subgroup lattice for $Z_{p^{2} q}$, where $p$ and $q$ are distinct primes.
Determine the subgroup lattice for $Z_{8}$.
Determine the subgroup lattice for $Z_{p^{n}}$, where $p$ is a prime and $n$ is some positive integer.
Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic.
Show that the group of positive rational numbers under multiplication is not cyclic.
Consider the set $\{4,8,12,16\} .$ Show that this set is a group under multiplication modulo 20 by constructing its Cayley table. What is the identity element? Is the group cyclic? If so, find all of its generators.
Give an example of a group that has exactly 6 subgroups (including the trivial subgroup and the group itself). Generalize to exactly $n$ subgroups for any positive integer $n .$
Let $m$ and $n$ be elements of the group $Z$. Find a generator for the $\operatorname{group}\langle m\rangle \cap\langle n\rangle .$
Suppose that $a$ and $b$ are group elements that commute and have orders $m$ and $n$. If $\langle a\rangle \cap\langle b\rangle=\{e\}$, prove that the group contains an element whose order is the least common multiple of $m$ and $n .$ Show that this need not be true if $a$ and $b$ do not commute.
Suppose that $a$ and $b$ belong to a group $G, a$ and $b$ commute, and $|a|$ and $|b|$ are finite. What are the possibilities for $|a b|$ ?
Suppose that $a$ and $b$ belong to a group $G, a$ and $b$ commute, and $|a|$ and $|b|$ are finite. Prove that $G$ has an element of order $\operatorname{lcm}(|a|,|b|)$.
Let $F$ and $F^{\prime}$ be distinct reflections in $D_{21} .$ What are the possibilities for $\left|F F^{\prime}\right| ?$
Suppose that $H$ is a subgroup of a group $G$ and $|H|=10 .$ If $a$ belongs to $G$ and $a^{6}$ belongs to $H$, what are the possibilities for $|a|$ ?
Which of the following numbers could be the exact number of elements of order 21 in a group: $21600,21602,21604 ?$
If $G$ is an infinite group, what can you say about the number of elements of order 8 in the group? Generalize.
Suppose that $K$ is a proper subgroup of $D_{35}$ and $K$ contains at least two reflections. What are the possible orders of $K ?$ Explain your reasoning.
For each positive integer $n$, prove that $C^{*}$, the group of nonzero complex numbers under multiplication, has exactly $\phi(n)$ elements of order $n$.
Prove or disprove that $H=\{n \in Z \mid n$ is divisible by both 8 and 10$\}$ is a subgroup of $Z$.
Suppose that $G$ is a finite group with the property that every nonidentity element has prime order (for example, $D_{3}$ and $D_{5}$ ). If $Z(G)$ is not trivial, prove that every nonidentity element of $G$ has the same order.
Prove that an infinite group must have an infinite number of subgroups.
Let $p$ be a prime. If a group has more than $p-1$ elements of order $p$, why can't the group be cyclic?
Suppose that $G$ is a cyclic group and that 6 divides $|G|$. How many elements of order 6 does $G$ have? If 8 divides $|G|$, how many elements of order 8 does $G$ have? If $a$ is one element of order 8 , list the other elements of order 8 .
List all the elements of $Z_{40}$ that have order $10 .$ Let $|x|=40$. List all the elements of $\langle x\rangle$ that have order 10 .
Reformulate the corollary of Theorem $4.4$ to include the case when the group has infinite order.
Determine the orders of the elements of $D_{33}$ and how many there are of each.
If $G$ is a cyclic group and 15 divides the order of $G$, determine the number of solutions in $G$ of the equation $x^{15}=e$. If 20 divides the order of $G$, determine the number of solutions of $x^{20}=e$. Generalize.
If $G$ is an Abelian group and contains cyclic subgroups of orders 4 and 5, what other sizes of cyclic subgroups must $G$ contain? Generalize.
If $G$ is an Abelian group and contains cyclic subgroups of orders 4 and 6, what other sizes of cyclic subgroups must $G$ contain? Generalize.
Prove that no group can have exactly two elements of order $2 .$
Given the fact that $U(49)$ is cyclic and has 42 elements, deduce the number of generators that $U(49)$ has without actually finding any of the generators.
$$\begin{aligned}&\text { Let } a \text { and } b \text { be elements of a group. If }|a|=10 \text { and }|b|=21 \text { , show }\\&\text { that }\langle a\rangle \cap\langle b\rangle=\{e\} \text { . }\end{aligned}$$
$$\begin{aligned}&\text { Let } a \text { and } b \text { belong to a group. If }|a| \text { and }|b| \text { are relatively prime, }\\&\text { show that }\langle a\rangle \cap\langle b\rangle=\{e\} \text { . }\end{aligned}$$
Let $a$ and $b$ belong to a group. If $|a|=24$ and $|b|=10$, what are the possibilities for $|\langle a\rangle \cap\langle b\rangle|$ ?
Prove that $U\left(2^{n}\right)(n \geq 3)$ is not cyclic.
Suppose that $G$ is a group of order 16 and that, by direct computation, you know that $G$ has at least nine elements $x$ such that $x^{8}=e .$ Can you conclude that $G$ is not cyclic? What if $G$ has at least five elements $x$ such that $x^{4}=e ?$ Generalize.
Prove that $Z_{n}$ has an even number of generators if $n>2 .$ What does this tell you about $\phi(n)$ ?
If $\left|a^{5}\right|=12$, what are the possibilities for $|a|$ ? If $\left|a^{4}\right|=12$, what are the possibilities for $|a|$ ?
Suppose that $|x|=n$. Find a necessary and sufficient condition on $r$ and $s$ such that $\left\langle x^{r}\right\rangle \subseteq\left\langle x^{s}\right\rangle$.
Suppose $a$ is a group element such that $\left|a^{28}\right|=10$ and $\left|a^{22}\right|=20$. Determine $|a| .$
Let $a$ be a group element such that $|a|=48$. For each part, find a divisor $k$ of 48 such thata. $\left\langle a^{21}\right\rangle=\left\langle a^{k}\right\rangle$;b. $\left\langle a^{14}\right\rangle=\left\langle a^{k}\right\rangle$;c. $\left\langle a^{18}\right\rangle=\left\langle a^{k}\right\rangle$.
Let $p$ be a prime. Show that in a cyclic group of order $p^{n}-1$, every element is a $p$ th power (that is, every element can be written in the form $a^{p}$ for some $a$ ).
Prove that $\left.H=\left\{\begin{array}{ll}1 & n \\ 0 & 1\end{array}\right] \mid n \in Z\right\}$ is a cyclic subgroup of
Let $a$ and $b$ belong to a group. If $|a|=12,|b|=22$, and $\langle a\rangle \cap\langle b\rangle \neq$ $\{e\}$, prove that $a^{6}=b^{11}$.
(2008 GRE Practice Exam) If $x$ is an element of a cyclic group of order 15 and exactly two of $x^{3}, x^{5}$, and $x^{9}$ are equal, determine $\left|x^{13}\right|$.
Determine the number of cyclic subgroups of order 4 in $D_{n}$.
If $n$ is odd, prove that $D_{n}$ has no subgroup of order 4 .
If $n \geq 4$ and is even, show that $D_{n}$ has exactly $n / 2$ noncyclic subgroups of order 4 .
If $n \geq 4$ and $n$ is divisible by 2 but not by 4, prove that $D_{n}$ has exactly $n / 2$ subgroups of order $4 .$
How many subgroups of order $n$ does $D_{n}$ have?
Let $G$ be the set of all polynomials of the form $a x^{2}+b x+c$ with coefficients from the set $\{0,1,2\} .$ We can make $G$ a group under addition by adding the polynomials in the usual way, except that we use modulo 3 to combine the coefficients. With this operation, prove that $G$ is a group of order 27 that is not cyclic.
Let $a$ and $b$ belong to some group. Suppose that $|a|=m,|b|=n$, and $m$ and $n$ are relatively prime. If $a^{k}=b^{k}$ for some integer $k$, prove that $m n$ divides $k$.
For every integer $n$ greater than 2, prove that the group $U\left(n^{2}-1\right)$ is not cyclic.
Prove that for any prime $p$ and positive integer $n, \phi\left(p^{n}\right)=$ $p^{n}-p^{n-1}$
Give an example of an infinite group that has exactly two elements of order 4