Chapter Questions
Which of the following binary operations are closed?a. subtraction of positive integersb. division of nonzero integersc. function composition of polynomials with real coefficientsd. multiplication of $2 \times 2$ matrices with integer entriese. exponentiation of integers
Which of the following binary operations are associative?a. subtraction of integersb. division of nonzero rationalsc. function composition of polynomials with real coefficientsd. multiplication of $2 \times 2$ matrices with integer entriese. exponentiation of integers
Which of the following binary operations are commutative?a. substraction of integersb. division of nonzero real numbersc. function composition of polynomials with real coefficientsd. multiplication of $2 \times 2$ matrices with real entriese. exponentiation of integers
Which of the following sets are closed under the given operation?a. $\{0,4,8,12\}$ addition $\bmod 16$b. $\{0,4,8,12\}$ addition mod 15c. $\{1,4,7,13\}$ multiplication mod 15d. $\{1,4,5,7\}$ multiplication mod 9
In each case, find the inverse of the element under the given operation.a. 13 in $Z_{20}$b. 13 in $U(14)$c. $n-1$ in $U(n)(n>2)$d. $3-2 i$ in $\mathbf{C}^{*}$, the group of nonzero complex numbers under multiplication
In each case, perform the indicated operation.a. $\operatorname{In} C^{*},(7+5 i)(-3+2 i)$b. $\operatorname{In} G L\left(2, Z_{13}\right), \operatorname{det}\left[\begin{array}{ll}7 & 4 \\ 1 & 5\end{array}\right]$c. $\operatorname{In} G L(2, \mathrm{R}),\left[\begin{array}{ll}6 & 3 \\ 8 & 2\end{array}\right]^{-1}$d. In $G L\left(2, Z_{7}\right),\left[\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right]^{-1}$
Give two reasons why the set of odd integers under addition is not a group.
List the elements of $U(20)$.
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 $G L(2, \mathbf{R})$ of Example 9 is non-Abelian by exhibiting a pair of matrices $A$ and $B$ in $G L(2, \mathbf{R})$ such that $A B \neq B A$.
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 $k$.
In $U(9)$ find the inverse of 2,7, and 8 .
Translate each of the following multiplicative expressions into its additive counterpart. Assume that the operation is commutative.a. $a^{2} b^{3}$b. $a^{-2}\left(b^{-1} c\right)^{2}$c. $\left(a b^{2}\right)^{-3} c^{2}=e$
For group elements $a, b$, and $c$, express $(a b)^{3}$ and $\left(a b^{-2} c\right)^{-2}$ 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 $\left(a^{2} b^{4}\right)^{-2}$ without using negative exponents.
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 $U(8)$ ?
Let $G$ be a group and let $H=\left\{x^{-1} \mid x \in G\right\} .$ Show that $G=H$ as sets.
List the members of $K=\left\{x^{2} \mid x \in D_{4}\right\}$ and $L=\left\{x \in D_{4} \mid x^{2}=e\right\}$.
Prove that the set of all $2 \times 2$ matrices with entries from $\mathbf{R}$ 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, $x y=z x$ implies $y=z$. Prove that $G$ is Abelian. ("Left-right cancellation" implies commutativity.)
(Law of Exponents for Abelian Groups) Let $a$ and $b$ be elements of an Abelian group and let $n$ be any integer. Show that $(a b)^{n}=a^{n} b^{n}$. Is this also true for non-Abelian groups?
(Socks-Shoes Property) Draw an analogy between the statement $(a b)^{-1}=b^{-1} a^{-1}$ 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 $(a b)^{-1}=a^{-1} b^{-1}$. Find an example that shows that in a group, it is possible to have $(a b)^{-2} \neq b^{-2} a^{-2}$. What would be an appropriate name for the group property $(a b c)^{-1}=c^{-1} b^{-1} a^{-1} ?$
Prove that a group $G$ is Abelian if and only if $(a b)^{-1}=a^{-1} b^{-1}$ for all $a$ and $b$ in $G$.
Prove that in a group, $\left(a^{-1}\right)^{-1}=a$ for all $a$.
For any elements $a$ and $b$ from a group and any integer $n$, prove that $\left(a^{-1} b a\right)^{n}=a^{-1} b^{n} a$
If $a_{1}, a_{2}, \ldots, a_{n}$ belong to a group, what is the inverse of $a_{1} a_{2} \cdots a_{n}$ ?
The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication modulo 56 . List all 12 .
Give an example of a group with 105 elements. Give two examples of groups with 44 elements.
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.
Construct a Cayley table for $U(12)$.
Suppose the table below is a group table. Fill in the blank entries.
Prove that in a group, $(a b)^{2}=a^{2} b^{2}$ if and only if $a b=b a$. Prove that in a group, $(a b)^{-2}=b^{-2} a^{-2}$ if and only if $a b=b a$.
Let $a, b$, and $c$ be elements of a group. Solve the equation $a x b=c$ for $x$. Solve $a^{-1} x a=c$ for $x$.
Let $a$ and $b$ belong to a group $G$. Find an $x$ in $G$ such that $x a b x^{-1}=b a$.
Let $G$ be a finite group. Show that the number of elements $x$ of $G$ such that $x^{3}=e$ is odd. Show that the number of elements $x$ of $G$ such that $x^{2} \neq e$ is even.
Give an example of a group with elements $a, b, c, d$, and $x$ such that $a x b=c x d$ but $a b \neq c d$. (Hence "middle cancellation" is not valid in groups.)
Suppose that $G$ is a group with the property that for every choice of elements in $G, a x b=c x d$ implies $a b=c d$. Prove that $G$ is Abelian. ("Middle cancellation" implies commutativity.)
Find an element $X$ in $D_{4}$ such that $R_{90} V X H=D^{\prime}$.
Suppose $F_{1}$ and $F_{2}$ are distinct reflections in a dihedral group $D_{n}$. Prove that $F_{1} F_{2} \neq R_{0}$.
Suppose $F_{1}$ and $F_{2}$ are distinct reflections in a dihedral group $D_{n}$ such that $F_{1} F_{2}=F_{2} F_{1}$. Prove that $F_{1} F_{2}=R_{180}$.
Let $R$ be any fixed rotation and $F$ any fixed reflection in a dihedral group. Prove that $R^{k} F R^{k}=F$.
Let $R$ be any fixed rotation and $F$ any fixed reflection in a dihedral group. Prove that $F R^{k} F=R^{-k}$. Why does this imply that $D_{n}$ is non-Abelian?
In the dihedral group $D_{n}$, let $R=R_{360 / n}$ and let $F$ be any reflection. Write each of the following products in the form $R^{i}$ or $R^{i} F$, where $0 \leq i<n$a. In $D_{4}, F R^{-2} F R^{5}$b. In $D_{5}^{4}, R^{-3} F R^{4} F R^{-2}$c. $\operatorname{In} D_{6}, F R^{5} F R^{-2} F$
Prove that the set of all $3 \times 3$ matrices with real entries of the form$$\left[\begin{array}{lll}1 & a & b \\0 & 1 & c \\0 & 0 & 1\end{array}\right]$$is a group. (Multiplication is defined by$$\left[\begin{array}{lll}1 & a & b \\0 & 1 & c \\0 & 0 & 1\end{array}\right]\left[\begin{array}{ccc}1 & a^{\prime} & b^{\prime} \\0 & 1 & c^{\prime} \\0 & 0 & 1\end{array}\right]=\left[\begin{array}{ccc}1 & a+a^{\prime} & b^{\prime}+a c^{\prime}+b \\0 & 1 & c^{\prime}+c \\0 & 0 & 1\end{array}\right]$$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 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 26.)
In a finite group, show that the number of nonidentity elements that satisfy the equation $x^{5}=e$ is a multiple of 5 . 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 $\mathrm{GL}\left(2, Z_{2}\right)$. Show that this group is nonAbelian by finding two elements that do not commute. (This exercise is referred to in Chapter 7.)
Prove the assertion made in Example 19 that the set $\{1,2, \ldots,$, $n-1\}$ is a group under multiplication modulo $n$ if and only if $n$ is prime.
Suppose that in the definition of a group $G$, the condition that there exists an element $e$ with the property $a e=e a=a$ for all $a$ in $G$ is replaced by $a e=a$ for all $a$ in $G$. Show that $e a=a$ for all $a$ in $G$. (Thus, a one-sided identity is a two-sided identity.)
Suppose that in the definition of a group $G$, the condition that for each element $a$ in $G$ there exists an element $b$ in $G$ with the property $a b=b a=e$ is replaced by the condition $a b=e$. Show that $b a=e$. (Thus, a one-sided inverse is a two-sided inverse.)