Finite Groups; Subgroups

For each group in the following list, find the order of the group and the order of each element in the group. What relation do you see between the orders of the elements of a group and the order of the group?

Let $\mathbb {Q}$ be the group of rational numbers under addition and let $\mathbb {Q}^*$ be the group of nonzero rational numbers under multiplication. In $\mathbb {Q}$, list the elements in $\left\langle \frac{1}{2}\right\rangle$. In $\mathbb {Q}^*$, list the elements in $\left\langle \frac{1}{2}\right\rangle$.

Let $\mathbb {Q}$ and $\mathbb {Q}^*$ be as in Exercise 3.2. Find the order of each element in $\mathbb {Q}$ and in $\mathbb {Q}^*$.

For every group element $a$, prove that $\left|a\right| = \left|a^{-1}\right|$.

Let $a$ belong to a group and $\left|a\right| = m$. If $n$ is relatively prime to $m$, show that $a$ can be written as the $n$th power of some element in the group.

Give an example of a group $G$ under addition that has a nonempty subset $H$ that is closed but is not a subgroup of $G$. Do the same for a group $G$ under multiplication.

Without actually computing the orders, explain why the two elements in each of the following pairs of elements from $\mathbb {Z}_{30}$ must have the same order: $\{ 2, 28\}$, $\{ 8, 22\}$. Do the same for the following pairs of elements from $U(15)$: $\{ 2, 8\}$, $\{ 7, 13\}$.

In the group $\mathbb {Z}_{12}$, find $\left|a\right|$, $\left|b\right|$, and $\left|a+b\right|$ for each case.

If $a$, $b$, and $c$ are group elements and $\left|a\right| = 6$, $\left|b\right| = 7$, express $(a^4c^{-2}b^4)^{-1}$ without using negative exponents.

Given that $\{ e, a, a^2, b, ab, a^2b\}$ is a group where $\left|a\right| = 3$, $\left|b\right| = 2$ and $ba = a^2b$, determine which of the six elements is $aba^2$ and which is $a^2bab$.

Find a non-Abelian subgroup of order $6$ in $D_6$.

Is $K = \{ x^2 \mid x \in D_4\}$ a subgroup of $D_4$? Is $K = \{ x^3 \mid x \in D_3\}$ a subgroup of $D_3$? What about $D_6$?

What can you say about a subgroup of $D_4$ that contains $R_{270}$ and a reflection? What can you say about a subgroup of $D_4$ that contains $H$ and $D$? What can you say about a subgroup of $D_4$ that contains $H$ and $V$?

How many subgroups of order $4$ does $D_4$ have?

Determine all elements of finite order in $\mathbb {R}^*$, the group of nonzero real numbers under multiplication.

Complete the statement ``A group element $x$ is its own inverse if and only if $\left|x\right| =$ .''

For any group elements $a$ and $x$, prove that $\left|xax^{-1}\right| = \left|a\right|$. This exercise is referred to in Chapter 23.

Prove that if $a$ is the only element of order $2$ in a group, then $a$ lies in the center of the group.

Prove that no group is the union of two proper subgroups. Does the statement remain true if two'' is replaced by three''?

Let $G$ be a group and $n$ an odd positive integer with the property that $x^n = e$ for all $x$ in $G$. If $a$ and $b$ belong to $G$ and $a^2 = b^2$ prove that $a = b$.

For each divisor $k > 1$ of $n$, let $U_k(n) = \{ x \in U(n) \mid x \bmod k = 1\}$. List the elements of $U_4(20)$, $U_5(20)$, $U_5(30)$, and $U_{10}(30)$. Prove that $U_k(n)$ is a subgroup of $U(n)$. (This exercise is referred to in Chapter 8.)

Suppose that $a$ is a group element and $a^6 = e$. What are the possibilities for $\left|a\right|$? Provide reasons for your answer.

If $a$ is a group element and $a$ has infinite order, prove that $a^m \neq a^n$ when $m \neq n$.

For any group elements $a$ and $b$, prove that $\left|ab\right| = \left|ba\right|$. Explain why this proves that $\left|abab\right| = \left|baba\right|$. Is it true that $\left|aba\right| = \left|bab\right|$?

If $A$ belongs to $GL(2, \mathbb {R})$ and $\left|A\right|$ is finite, what can you say about $\det A$?

Let $a$ and $b$ be elements of a group and $\left|b\right| = 4$. If $a^n = b$, find a positive integer $k$ such that $a^{-1} = a^k$.

Show that $U(14) = \left\langle 3\right\rangle = \left\langle 5\right\rangle$. Is $U(14) = \left\langle 11\right\rangle$? Show that $U(20) \neq \left\langle k\right\rangle$ for any $k$ in $U(20)$.

Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb {Z}_n$. Prove that either every member of $H$ is even or exactly half of the members of $H$ are even.

Let $n$ be a positive even integer and let $H$ be a subgroup of $\mathbb {Z}_n$ of odd order. Prove that every member of $H$ is an even integer.

Prove that for every subgroup of $D_n$, either every member of the subgroup is a rotation or exactly half of the members are rotations.

Let $H$ be a subgroup of $D_n$ of odd order. Prove that every member of $H$ is a rotation.

Prove that a group with two elements of order $2$ that commute must have a subgroup of order $4$.

For every even integer $n$, show that $D_n$ has a subgroup of order $4$.

Suppose that $H$ is a proper subgroup of $\mathbb {Z}$ under addition and $H$ contains $18$, $30$, and $40$. Determine $H$.

Suppose that $H$ is a proper subgroup of $\mathbb {Z}$ under addition and that $H$ contains $12$, $30$, and $54$. What are the possibilities for $H$?

Suppose that $H$ is a subgroup of $\mathbb {Z}$ under addition and that $H$ contains $250$ and $350$. What are the possibilities for $H$?

Prove that the dihedral group of order $6$ does not have a subgroup of order $4$.

If $H$ and $K$ are subgroups of $G$, show that $H \cap K$ is a subgroup of $G$. (Can you see that the same proof shows that the intersection of any number of subgroups of $G$, finite or infinite, is again a subgroup of $G$?)

Let $H_1$ and $H_2$ be subgroups of $D_n$ of order $4$. What subgroup is $H_1 \cap H_2$?

How many subgroups of order $6$ does $D_6$ have? How many does $D_{12}$ have? Generalize to $D_n$ where $n$ is a positive integer divisible by $6$.

Let $G$ be a group. Show that $Z(G) = \bigcap_{a \in G} C(a)$.

Recall $Z(G) = \{ a \in G \mid ax = xa \text{ for all } x \text{ in } G\}$ is the center of $G$, and for $a \in G$, the centralizer $C(a) = \{ g \in G \mid ga = ag\}$ is the set of all elements that commute with $a$.

Let $G$ be a group, and let $a \in G$. Prove that $C(a) = C(a^{-1})$.

For any group element $a$ and any integer $k$, show that $C(a) \subseteq C(a^k)$. Use this fact to complete the following statement: ``In a group, if $x$ commutes with $a$, then .'' Is the converse true?

If $a$ and $b$ are distinct group elements, prove that either $a^2 \neq b^2$ or $a^3 \neq b^3$.

Let $S$ be a subset of a group and let $H$ be the intersection of all subgroups of $G$ that contain $S$.

In the group $\mathbb {Z}$, find

Prove Theorem 3.6.

Theorem 3.6 (C(a) Is a Subgroup) states: for each $a$ in a group $G$, the centralizer of $a$, $C(a) = \{ g \in G \mid ga = ag\}$, is a subgroup of $G$.

If $H$ is a subgroup of $G$, then by the centralizer $C(H)$ of $H$ we mean the set $\{ x \in G \mid xh = hx \text{ for all } h \in H\}$. Prove that $C(H)$ is a subgroup of $G$.

Must the centralizer of an element of a group be Abelian? Must the center of a group be Abelian?

Suppose $a$ belongs to a group and $\left|a\right| = 5$. Prove that $C(a) = C(a^3)$. Find an element $a$ from some group such that $\left|a\right| = 6$ and $C(a) \neq C(a^3)$.

Let $G$ be an Abelian group with identity $e$ and let $n$ be some fixed integer. Prove that the set of all elements of $G$ that satisfy the equation $x^n = e$ is a subgroup of $G$. Give an example of a group $G$ in which the set of all elements of $G$ that satisfy the equation $x^2 = e$ does not form a subgroup of $G$. (This exercise is referred to in Chapter 11.)

In each case, find elements $a$ and $b$ from a group such that $\left|a\right| = \left|b\right| = 2$ and

Consider the element $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ in $SL(2, \mathbb {R})$. What is the order of $A$? If we view $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ as a member of $SL(2, \mathbb {Z}_p)$ ($p$ is a prime), what is the order of $A$?

Consider the elements $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ -1 & -1 \end{bmatrix}$ from $SL(2, \mathbb {R})$. Find $\left|A\right|$, $\left|B\right|$, and $\left|AB\right|$. Does your answer surprise you?

Let $a$ be a group element of order $n$, and suppose that $d$ is a positive divisor of $n$. Prove that $\left|a^d\right| = n/d$.

Give an example of elements $a$ and $b$ from a group such that $a$ has finite order, $b$ has infinite order and $ab$ has finite order.

Prove that a group of even order must have an odd number of elements of order $2$.

For any positive integer $n$ and any angle $\theta$, show that in the group $SL(2, \mathbb {R})$,

Use this formula to find the order of

(Geometrically, $\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ represents a rotation of the plane $\theta$ degrees.)

Let $G$ be the symmetry group of a circle. Show that $G$ has elements of every finite order as well as elements of infinite order.

In the group $\mathbb {R}^*$ find elements $a$ and $b$ such that $\left|a\right| = \infty$, $\left|b\right| = \infty$ and $\left|ab\right| = 2$.

Let $G$ be the symmetry group of a circle. Explain why $G$ contains $D_n$ for all $n$.

Let $G$ be the group of symmetries of a circle. Use the fact from plane geometry that if $F$ and $F'$ are reflections that intersect in an angle of $\theta$ degrees then $FF'$ is a rotation of $2\theta$ degrees to find reflections $F$ and $F'$ such that $FF'$ has infinite order.

Let $H$ be a subgroup of a finite group $G$. Suppose that $g$ belongs to $G$ and $n$ is the smallest positive integer such that $g^n \in H$. Prove that $n$ divides $\left|g\right|$.

Compute the orders of the following groups.

Let $\mathbb {R}^*$ be the group of nonzero real numbers under multiplication and let $H = \{ x \in \mathbb {R}^* \mid x^2 \text{ is rational}\}$. Given an example of an irrational number that is in $H$. Prove that $H$ is a subgroup of $\mathbb {R}^*$. Can the exponent $2$ be replaced by any positive integer and still have $H$ be a subgroup? If $H = \{ x \in \mathbb {R}^* \mid x^n \text{ is rational for some integer } n\}$ ($n$ may vary with $x$) does your proof that $H$ is a subgroup remain valid?

Find a noncyclic subgroup of order $4$ in $U(40)$.

Suppose that $a$ is an element of a group and $a^m = e$. Prove that $\left|a\right|$ divides $m$.

Let $G$ be an Abelian group and $H = \{ x \in G \mid x^n = e \text{ for some odd integer } n\}$ ($n$ may vary with $x$). Prove that $H$ is a subgroup of $G$.

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

Prove that $H = \{ x \in \mathbb {C}^* \mid x^n = \pm 1 \text{ for some integer } n\}$ ($n$ may vary with $x$) is a subgroup of the nonzero complex numbers under multiplication. Let $K$ be any subgroup of $\mathbb {C}^*$. Does your argument work when the condition $x^n = \pm 1$ is replaced by $x^n \in K$.

Let $G$ be a group and let $H$ be a subgroup of $G$. For any fixed $x$ in $G$, define $xHx^{-1} = \{ xhx^{-1} \mid h \in H\}$. Prove the following:

Let $G = GL(2, \mathbb {R})$ and $H = \{ A \in G \mid \det A \text{ is a positive rational number}\}$. Prove that $H$ is a subgroup of $G$.

Show that the set of all rational numbers that, when expressed in lowest terms, have an odd numerator and an odd denominator is a subgroup of $\mathbb {Q}^*$ under multiplication.

Let $H = \{ A \in GL(2, \mathbb {R}) \mid \det A$ is an integer power of $2\}$. Show that $H$ is a subgroup of $GL(2, \mathbb {R})$.

Let $H$ be a subgroup of $\mathbb {R}$ under addition. Let $K = \{ 2^a \mid a \in H\}$. Prove that $K$ is a subgroup of $\mathbb {R}^*$ under multiplication.

Let $G$ be a group of functions from $\mathbb {R}^*$ to $\mathbb {R}^*$, where the operation of $G$ is multiplication of functions. Let $H = \{ f \in G \mid f(2) = 1\}$. Prove that $H$ is a subgroup of $G$. Can $2$ be replaced by any real number?

Let $G = GL(2, \mathbb {R})$ and $H = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \mid a \text{ and } b \text{ are nonzero integers}\right\}$ under the operation of matrix multiplication. Prove or disprove that $H$ is a subgroup of $GL(2, \mathbb {R})$.

For any odd integer $n > 3$ give two examples of groups with exactly $n$ elements of order $2$.

Let $H = \{ a + bi \mid a, b \in \mathbb {R}, a^2 + b^2 = 1\}$. Prove or disprove that $H$ is a subgroup of $\mathbb {C}^*$ under multiplication. Describe the elements of $H$ geometrically.

Let $G$ be a finite Abelian group and let $a$ and $b$ belong to $G$. Prove that the set $\left\langle a,b\right\rangle = \{ a^ib^j \mid i,j \in \mathbb {Z}\}$ is a subgroup of $G$. What can you say about $\left| \left\langle a,b\right\rangle \right|$ in terms of $\left|a\right|$ and $\left|b\right|$?

Let $n$ be an integer greater than $1$. Find a subgroup of $U(4n)$ of order $4$. Is your subgroup cyclic?

Give an example to show that if $H$ and $K$ are subgroups of a group $G$ the set $HK = \{ hk \mid h \in H, k \in K\}$ need not be a subgroup of $G$.

Find an example of a group with elements $a$ and $b$ such that $\left|a\right| = 2$, $\left|b\right| = 10$, and $\left|ab\right| = 2$.

If $p$ is an odd prime and $n$ is a positive integer prove that $p^n - 1$ is the unique element in $U(p^n)$ of order $2$.

If $n$ is an integer at least $3$, prove that $2^n - 1$ and $2^{n-1} \pm 1$ are the only elements in $U(2^n)$ of order $2$.

For parts a and b below give examples of a finite group and an infinite group for which each set is a proper subgroup. For parts c and d give finite examples.

Let $H$ be a subgroup of a group $G$. Prove that the set $HZ(G) = \{ hz \mid h \in H, z \in Z(G)\}$ is a subgroup of $G$. This exercise is referred to in this chapter.

If $H$ and $K$ are nontrivial subgroups of the rational numbers under addition, prove that $H \cap K$ is nontrivial.

Let $H$ be a nontrivial subgroup of the group of rational numbers under addition. Prove that $H$ has a nontrivial proper subgroup.

Prove that a group of order $n$ greater than $2$ cannot have a subgroup of order $n - 1$.

If $G$ is a finite group and $x$ is an element of $G$ give a probabilistic interpretation of $\left|C(x)\right|/\left|G\right|$. Prove that the probability that any two randomly chosen elements (they can be the same) from $D_4$ commute is greater than $.5$. What is the probability that any two elements from $D_8$ commute?

Let $G$ be a finite group with more than one element. Show that $G$ has an element of prime order.