Groups

Which of the following binary operations are closed?

Which of the following binary operations are associative?

Which of the following binary operations are commutative?

Which of the following sets are closed under the given operation?

In each case, find the inverse of the element under the given operation.

In each case, perform the indicated operation.

Which of the following sets are groups under matrix multiplication?

Does the set of integers under subtraction (i.e., $a - b$) have an identity element?

Show that the equation $5x = 3$ modulo $20$ has no integer solution but $3x = 5$ does.

List the elements of $U(20)$ and find the inverse of each one.

For group elements $a$, $b$, and $c$ provide a descriptive name for the property $(abc)^{-1} = c^{-1}b^{-1}a^{-1}$.

Give two reasons why the set of odd integers under addition is not a group.

Show that $\{ 1, 2, 3\}$ under multiplication modulo $4$ is not a group but that $\{ 1, 2, 3, 4\}$ under multiplication modulo $5$ is a group.

Show that the group $GL(2, \mathbb {R})$ of Example 9 is non-Abelian by exhibiting a pair of matrices $A$ and $B$ in $GL(2, \mathbb {R})$ such that $AB \neq BA$.

Let $a$ belong to a group and $a^{12} = e$. Express the inverse of each of the elements $a, a^6, a^8$, and $a^{11}$ in the form $a^k$ for some positive integer $k$.

Show that the set $\{ 5, 15, 25, 35\}$ is a group under multiplication modulo $40$. What is the identity element of this group? Can you see any relationship between this group and $U(8)$?

Translate each of the following multiplicative expressions into its additive counterpart. Assume that the operation is commutative.

For group elements $a$, $b$, and $c$, express $(ab)^3$ and $(ab^{-2}c)^{-2}$ without parentheses.

Suppose that $a$ and $b$ belong to a group and $a^5 = e$ and $b^7 = e$. Write $a^{-2}b^{-4}$ and $(a^2b^4)^{-2}$ without using negative exponents.

List the members of $K = \{ x^2 \mid x \in D_4\}$ and $L = \{ x \in D_4 \mid x^2 = e\}$.

Prove that the set of all $2 \times 2$ matrices with entries from $\mathbb {R}$ and determinant $+1$ is a group under matrix multiplication.

For any integer $n > 2$, show that there are at least two elements in $U(n)$ that satisfy $x^2 = 1$.

An abstract algebra teacher intended to give a typist a list of nine integers that form a group under multiplication modulo $91$. Instead, one of the nine integers was inadvertently left out, so that the list appeared as $1, 9, 16, 22, 53, 74, 79, 81$. Which integer was left out? (This really happened!)

Let $G$ be a group with the property that for any $x, y, z$ in the group, $xy = zx$ implies $y = z$. Prove that $G$ is Abelian. (``Left-right cancellation'' implies commutativity.)

Let $a$ and $b$ be elements of an Abelian group and let $n$ be any integer. Show that $(ab)^n = a^n b^n$. Is this also true for non-Abelian groups?

Draw an analogy between the statement $(ab)^{-1} = b^{-1}a^{-1}$ and the act of putting on and taking off your socks and shoes. Find distinct nonidentity elements $a$ and $b$ from a non-Abelian group such that $(ab)^{-1} = a^{-1}b^{-1}$. Find an example that shows that in a group, it is possible to have $(ab)^{-2} \neq b^{-2}a^{-2}$. What would be an appropriate name for the group property $(abc)^{-1} = c^{-1}b^{-1}a^{-1}$?

Prove that a group $G$ is Abelian if and only if $(ab)^{-1} = a^{-1}b^{-1}$ for all $a$ and $b$ in $G$.

Prove that in a group, $(a^{-1})^{-1} = a$ for all $a$.

For any elements $a$ and $b$ from a group and any integer $n$, prove that $(a^{-1}ba)^n = a^{-1}b^na$.

If $a_1, a_2, \ldots , a_n$ belong to a group, what is the inverse of $a_1a_2 \ldots a_n$?

The integers $5$ and $15$ are among a collection of $12$ integers that form a group under multiplication modulo $56$. List all $12$.

For a positive integer $n$ let $S_n$ be the set of all 1-1 functions from $\{ 1, 2, \ldots , n\}$ to itself. What is the identity under the operation of function composition? Does every element of $S_n$ have an inverse in $S_n$ under this operation? Is $S_n$ closed under this operation? Is $S_n$ a group under this operation? How many elements are in $S_n$?

Prove that every group table is a Latin square; that is, each element of the group appears exactly once in each row and each column.

Give an example of a group with exactly $105$ elements. Give three examples of groups with $40$ elements.

Prove that the set of all $3 \times 3$ matrices with real entries of the form

is a group. (Multiplication is defined by

This group, sometimes called the Heisenberg group after the Nobel Prize--winning physicist Werner Heisenberg, is intimately related to the Heisenberg Uncertainty Principle of quantum physics.)

Prove that in a group, $(ab)^2 = a^2b^2$ if and only if $ab = ba$. Prove that in a group, $(ab)^{-2} = b^{-2}a^{-2}$ if and only if $ab = ba$.

Let $G$ be a finite group and $n$ an odd positive integer. Show that the number of elements $x$ of $G$ such that $x^n = e$ is odd. Show that the number of elements $x$ of $G$ such that $x^2 \neq e$ is even.

Let $\mathbb {Q}^+$ denote the positive rational numbers. Explain why the correspondence from $\mathbb {Q}^+ \times \mathbb {Q}^+$ that takes $\left(\frac{a}{b}, \frac{c}{d}\right)$ to $\frac{a+c}{b+d}$ is not a binary operation on $\mathbb {Q}^+$. (That is, the correspondence is not well-defined.)

Suppose $F_1$ and $F_2$ are distinct reflections in a dihedral group $D_n$. Prove that $F_1F_2 \neq R_0$.

Suppose $F_1$ and $F_2$ are distinct reflections in a dihedral group $D_n$ such that $F_1F_2 = F_2F_1$. Prove that $F_1F_2 = R_{180}$.

Let $R$ be any fixed rotation and $F$ any fixed reflection in a dihedral group. Prove that $R^kFR^k = F$.

Let $R$ be any fixed rotation and $F$ any fixed reflection in a dihedral group. Prove that $FR^kF = R^{-k}$. Why does this imply that $D_n$ is non-Abelian?

Let $R$ be some rotation in $D_n$ and $F$ some reflection in $D_n$. Express $R^6FRFR^{-3}FRF$ in the form $R^k$. Express $FR^4FR^5FR^2$ in the form $R^kF$.

Let $R_\alpha$ and $R_\beta$ be rotations in $D_n$ and $F$ be a reflection in $D_n$. What rotation is $FR_\alpha F R_\beta$? What rotation is $R_\alpha F R_\beta F$?

Prove that if $G$ is a group with the property that the square of every element is the identity, then $G$ is Abelian. (This exercise is referred to in Chapter 25.)

In a finite group, show that the number of nonidentity elements that satisfy the equation $x^5 = e$ is a multiple of $4$. If the stipulation that the group be finite is omitted, what can you say about the number of nonidentity elements that satisfy the equation $x^5 = e$?

List the six elements of $GL(2, \mathbb {Z}_2)$. Show that this group is non-Abelian by finding two elements that do not commute. (This exercise is referred to in Chapter 7.)

Prove the assertion made in Example 20 that the set $\{ 1, 2, \ldots , n-1\}$ is a group under multiplication modulo $n$ if and only if $n$ is prime.

Suppose the table below is a group table. Fill in the blank entries.