Cyclic Groups

Find all generators of $\mathbb {Z}_6$, $\mathbb {Z}_8$, and $\mathbb {Z}_{20}$.

Suppose that $\left\langle a\right\rangle$, $\left\langle b\right\rangle$, and $\left\langle c\right\rangle$ are cyclic groups of orders 6, 8, and 20, respectively. Find all generators of $\left\langle a\right\rangle$, $\left\langle b\right\rangle$, and $\left\langle c\right\rangle$.

List the elements of the subgroups $\left\langle 20\right\rangle$ and $\left\langle 10\right\rangle$ in $\mathbb {Z}_{30}$. 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 $\left\langle 3\right\rangle$ and $\left\langle 15\right\rangle$ in $\mathbb {Z}_{18}$. 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 $\left\langle 3\right\rangle$ and $\left\langle 7\right\rangle$ 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 $G$.

How many subgroups does $\mathbb {Z}_{20}$ have? List a generator for each of these subgroups. Suppose that $G = \left\langle a\right\rangle$ and $|a| = 20$. How many subgroups does $G$ have? List a generator for each of these subgroups.

In $\mathbb {Z}_{24}$, list all generators for the subgroup of order 8. Let $G = \left\langle a\right\rangle$ 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 $\left\langle a^{-1}\right\rangle = \left\langle a\right\rangle$.

In $\mathbb {Z}$, find all generators of the subgroup $\left\langle 3\right\rangle$. If $a$ has infinite order, find all generators of the subgroup $\left\langle a^3\right\rangle$.

In $\mathbb {Z}$, find a generator of the subgroup $\left\langle 10\right\rangle \cap \left\langle 12\right\rangle$. In general, what is a generator of the subgroup $\left\langle m\right\rangle \cap \left\langle n\right\rangle$? If $a$ is a group element of infinite order, what is a generator of the subgroup $\left\langle a^m\right\rangle \cap \left\langle a^n\right\rangle$?

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?

Let $G$ be an Abelian group and let $H = \left\{ g \in G \mid |g| \text{ divides } 12\right\}$. 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.

Let $a$ be an element of a group.

Let $a$ be a generator of a cyclic group $G$. If $|G| = n$ is odd and $\left\langle a^6\right\rangle$ is a proper subgroup of $\left\langle a\right\rangle$ determine $| \left\langle a^6\right\rangle |$.

Let $a$ be a generator of a cyclic group $G$. If $|G| = n$ and $\left\langle a^{10}\right\rangle = \left\langle a^5\right\rangle$ prove that $n$ is odd.

Let $n$ be an element of a group. If $|a^2| = 3$, what are the possibilities for $|a|$? What are they if $|a^2| = 4$?

Let $p$ be prime and $n$ a positive integer. How many cyclic subgroups does $D_{p^n}$ have? If $p$ and $q$ are distinct primes how many cyclic subgroups does $D_{pq}$ have?

Prove that if $G$ is a group with the property that the square of every element is the identity then $G$ is Abelian.

Determine $\phi (81)$, $\phi (60)$, $\phi (105)$ where $\phi$ is the Euler phi function.

Let $a$ and $b$ elements of a finite group and $a$ and $b$ commute. Prove that $|ab| = |a||b|$ if and only if $\gcd (|a|, |b|) = 1$.

Suppose that $G$ is a group with more than one element and $G$ has no proper, nontrivial subgroups. Prove that $|G|$ is prime. (Do not assume at the outset that $G$ is finite.)

Let $H$ be a subgroup of $D_n$ of odd order. Use Exercise 31 in Chapter 3 to prove that every subgroup of $H$ is cyclic.

Give infinitely many examples of Abelian groups that have exactly 2 elements of order 3. Do the same for infinitely many non-Abelian groups. Do your examples generalize to the case of exactly $p - 1$ elements of order $p$ where $p$ is an odd prime?

If a cyclic group has an element of infinite order, how many elements of finite order does it have?

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 $G$.

Prove that a group of order 3 must be cyclic.

Let $\mathbb {Z}$ denote the group of integers under addition. Is every subgroup of $\mathbb {Z}$ cyclic? Why? Describe all the subgroups of $\mathbb {Z}$. Let $a$ be a group element with infinite order. Describe all subgroups of $\left\langle a\right\rangle$.

For any element $a$ in any group $G$, prove that $\left\langle a\right\rangle$ 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 $\mathbb {Z}$. Let $a$ be a group element that has infinite order. Find all generators of $\left\langle a\right\rangle$.

Prove that $\mathbb {C}^*$, the group of nonzero complex numbers under multiplication, has a cyclic subgroup of order $n$ for every positive integer $n$.

Suppose that there are infinitely many subgroups $H_1, H_2, H_3, \ldots$ of a group $G$ such that $H_1 \subset H_2 \subset H_3 \subset \cdots$. Prove that $H = \bigcup_{i=1}^{\infty } H_i$ is a subgroup of $G$.

List all the elements of order 8 in $\mathbb {Z}_{8000000}$. How do you know your list is complete? Let $a$ be a group element such that $|a| = 8000000$. List all elements of order 8 in $\left\langle a\right\rangle$. How do you know your list is complete?

Suppose that $G$ is a group with more than one element. If the only subgroups of $G$ are $\{ e\}$ and $G$, prove that $G$ is cyclic and has prime order.

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 $\mathbb {Z}_{12}$. Generalize to $\mathbb {Z}_{p^2 q}$, where $p$ and $q$ are distinct primes.

Determine the subgroup lattice for $\mathbb {Z}_8$. Generalize to $\mathbb {Z}_{p^n}$, 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. Why does this prove that the group of nonzero rationals 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 7 subgroups (including the trivial subgroup and the group itself). Generalize to exactly $n$ subgroups for any positive integer $n$.

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 $|ab|$?

Suppose that $a$ and $b$ are group elements that commute. If $|a|$ is finite and $|b|$ infinite, prove that $|ab|$ has infinite order.

For a positive integer $k$ and an Abelian group $G$ prove that $G^k = \left\{ x^k \mid x \in G\right\}$ is a subgroup of $G$. If $G$ is Abelian and $|G| = 40$, prove that $G^3 = G$, $G^{15} = G^5$, and $G^{28} = G^4$. Generalize to the case that $|G| = n$ and $k$ is any positive integer.

Let $a$ belong to a group and $|a| = 100$. Find $|a^{98}|$ and $|a^{70}|$.

Let $F$ and $F'$ be distinct reflections in $D_{21}$. What are the possibilities for $|FF'|$?

Suppose that $H$ is a cyclic 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.

If $G$ is a cyclic group of order $n$, prove that for every element $a$ in $G$, $a^n = e$.

For each positive integer $n$, prove that $\mathbb {C}^*$, the group of nonzero complex numbers under multiplication, has exactly $\phi (n)$ elements of order $n$.

Prove or disprove that $H = \left\{ n \in \mathbb {Z} \mid n \text{ is divisible by both} 8 \text{ and } 10\right\}$ is a subgroup of $\mathbb {Z}$. What happens if divisible by both 8 and 10'' is changed to divisible by 8 or 10?''

Suppose that $G$ is a finite group with the property that every nonidentity element has prime order (e.g., $D_3$ 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 $\mathbb {Z}_{40}$ that have order 10. Let $|x| = 40$. List all the elements of $\left\langle x\right\rangle$ that have order 10.

Reformulate the corollary of Theorem 4.4 to include the case when the group has infinite order.

(Recall the corollary of Theorem 4.4 for finite groups: in a finite group, the number of elements of order $d$ is a multiple of $\phi (d)$.)

Determine the orders of the elements of $D_{33}$ and how many there are of each.

When checking to see if $\left\langle 2\right\rangle = U(25)$ explain why it is sufficient to check that $2^{10} \neq 1$ and $2^4 \neq 1$.

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.

Let $a$ and $b$ be elements of a group. If $|a| = 10$ and $|b| = 21$, show that $\left\langle a\right\rangle \cap \left\langle b\right\rangle = \{ e\}$.

Let $a$ and $b$ belong to a group. If $|a|$ and $|b|$ are relatively prime, show that $\left\langle a\right\rangle \cap \left\langle b\right\rangle = \{ e\}$.

Let $a$ and $b$ belong to a group. If $|a| = 24$ and $|b| = 10$, what are the possibilities for $| \left\langle a\right\rangle \cap \left\langle b\right\rangle |$?

Let $a$ and $b$ belong to a group. If $|a| = 12$, $|b| = 22$, and $\left\langle a\right\rangle \cap \left\langle b\right\rangle \neq \{ e\}$, prove that $a^6 = b^{11}$.

Let $G$ be an Abelian group of order 15. Prove that $G$ cannot have 14 elements of order 3.

Prove that $\mathbb {Z}_n$ has an even number of generators if $n > 2$. What does this tell you about $\phi (n)$?

If $|a^5| = 12$, what are the possibilities for $|a|$? If $|a^4| = 12$, what are the possibilities for $|a|$?

Suppose that $|x| = n$. Find a necessary and sufficient condition on $r$ and $s$ such that $\left\langle x^r\right\rangle \subseteq \left\langle x^s\right\rangle$.

Let $a$ be a group element such that $|a| = 48$. For each part, find a divisor $k$ of 48 such that

Find a cyclic subgroup of $D_{60}$ of order 4 and a non-cyclic subgroup of $D_{60}$ of order 4.

Let $G$ be an Abelian group and $H = \left\{ x \in G \mid |x| \text{ is odd}\right\}$. Prove that $H$ is a subgroup of $G$.

Let $G$ be an Abelian group and $H = \left\{ x \in G \mid |x| \text{ is 1 or even}\right\}$. Give an example to show that $H$ need not be a subgroup of $G$.

Let $G$ be a group and $H$ a subgroup of $G$. Prove that $HZ(G) = \left\{ hz \mid h \in H,\ z \in Z(G)\right\}$ is a subgroup of $G$.

For every integer $n$ greater than 2, prove that the group $U(n^2 - 1)$ is not cyclic.

For every integer $n > 2$, show that $U(n^2 - 1)$ has a subgroup of order 4.

Prove that $U(2^n)$ ($n \geq 3$) is not cyclic.

Prove that for any prime $p$ and positive integer $n$, $\phi (p^n) = p^n - p^{n-1}$.

Let $\phi$ be the Euler $\phi$-function. For any positive integer $n$ prove that $|U(|U(n)|)| = \phi (\phi (n))$.

Let $\phi$ be the Euler phi function. For each positive integer $n$ prove that $n = \sum \phi (d)$ where the sum ranges over all positive divisors $d$ of $n$.

If $n$ is an even integer prove that $\phi (2n) = 2\phi (n)$.

(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 $|x^{13}|$.