# Exercises1.4Exercises

##### Exercise1

Yes/No. For each of the following, write Y if the object described is a well-defined set; otherwise, write N. You do NOT need to provide explanations or show work for this problem.

1. $\{z \in \C \,:\, |z|=1\}$

2. $\{\epsilon \in \R^+\,:\, \epsilon \mbox{ is sufficiently small} \}$

3. $\{q\in \Q \,:\, q \mbox{ can be written with denominator } 4\}$

4. $\{n \in \Z\,:\, n^2 \lt 0\}$

Solution
##### Exercise2

List the elements in the following sets, writing your answers as sets.

Example: $\{z\in \C\,:\,z^4=1\}$ Solution: $\{\pm 1, \pm i\}$

1. $\{z\in \R\,:\, z^2=5\}$

2. $\{m \in \Z\,:\, mn=50 \mbox{ for some } n\in \Z\}$

3. $\{a,b,c\}\times \{1,d\}$

4. $P(\{a,b,c\})$

Solution
##### Exercise3

Let $S$ be a set with cardinality $n\in \N\text{.}$ Use the cardinalities of $P(\{a,b\})$ and $P(\{a,b,c\})$ to make a conjecture about the cardinality of $P(S)\text{.}$ You do not need to prove that your conjecture is correct (but you should try to ensure it is correct).

Solution
##### Exercise4

Let $f: \Z^2 \to \R$ be defined by $f(a,b)=ab\text{.}$ (Note: technically, we should write $f((a,b))\text{,}$ not $f(a,b)\text{,}$ since $f$ is being applied to an ordered pair, but this is one of those cases in which mathematicians abuse notation in the interest of concision.)

1. What are $f$'s domain, codomain, and range?

2. Prove or disprove each of the following statements. (Your proofs do not need to be long to be correct!)

1. $f$ is onto;

2. $f$ is 1-1;

3. $f$ is a bijection. (You may refer to parts (i) and (ii) for this part.)

3. Find the images of the element $(6,-2)$ and of the set $\Z^- \times \Z^-$ under $f\text{.}$ (Remember that the image of an element is an element, and the image of a set is a set.)

4. Find the preimage of $\{2,3\}$ under $f\text{.}$ (Remember that the preimage of a set is a set.)

Solution
##### Exercise5

Let $S\text{,}$ $T\text{,}$ and $U$ be sets, and let $f: S\to T$ and $g: T\to U$ be onto. Prove that $g \circ f$ is onto.

Solution
##### Exercise6

Let $A$ and $B$ be sets with $|A|=m\lt \infty$ and $|B|=n\lt \infty\text{.}$ Prove that $|A\times B|=mn\text{.}$

Solution

# Exercises2.2Exercises, Part I

##### Exercise1

For each of the following, write Y if the given “operation” is a well-defined binary operation on the given set; otherwise, write N. In each case in which it isn't a well-defined binary operation on the set, provide a brief explanation. You do not need to prove or explain anything in the cases in which it is a binary operation.

1. $+$ on $\C^*$

2. $*$ on $\R^+$ defined by $x*y=\log_x y$

3. $*$ on $\M_2(\R)$ defined by $A*B=AB^{-1}$

4. $*$ on $\Q^*$ defined by $z*w=z/w$

Solution
##### Exercise2

Define $*$ on $\Q$ by $p*q=pq+1\text{.}$ Prove or disprove that $*$ is (a) commutative; (b) associative.

Solution
##### Exercise3

Prove that matrix multiplication is not commutative on $\M_2(\R)\text{.}$

Solution
##### Exercise4

Prove or disprove each of the following statements.

1. The set $2\Z=\{2x\,:\,x\in \Z\}$ is closed under addition in $\Z\text{.}$

2. The set $S=\{1,2,3\}$ is closed under multiplication in $\R\text{.}$

3. The set

\begin{equation*} U=\left\{ \begin{bmatrix} a \amp b\\ 0 \amp c \end{bmatrix}\,:\,a,b,c\in \R\right\} \end{equation*}

is closed under multiplication in $\M_2(\R)\text{.}$ (Recall that $U$ is the set of upper-triangular matrices in $\M_2(\R)\text{.}$)

Solution
##### Exercise5

Let $*$ be an associative and commutative binary operation on a set $S\text{.}$ An element $u\in S$ is said to be an idempotent in $S$ if $u*u=u\text{.}$ Let $H$ be the set of all idempotents in $S\text{.}$ Prove that $H$ is closed under $*\text{.}$

Solution

# Exercises2.8Exercises, Part II

##### Exercise1

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem.

1. For every positive integer $n\text{,}$ there exists a group of order $n\text{.}$

2. For every integer $n\geq 2\text{,}$ $\Z_n$ is abelian.

3. Every abelian group is finite.

4. For every integer $m$ and integer $n\geq 2\text{,}$ there exist infinitely many integers $a$ such that $a$ is congruent to $m$ modulo $n\text{.}$

5. A binary operation $*$ on a set $S$ is commutative if and only if there exist $a,b\in S$ such that $a*b=b*a\text{.}$

6. If $\langle S, *\rangle$ is a binary structure, then the elements of $S$ must be numbers.

7. If $e$ is an identity element of a binary structure (not necessarily a group) $\langle S,*\rangle\text{,}$ then $e$ is an idempotent in $S$ (that is, $e*e=e$).

8. If $s$ is an idempotent in a binary structure (not necessarily a group) $\langle S,*\rangle\text{,}$ then $s$ must be an identity element of $S\text{.}$

Solution
##### Exercise2

Let $G$ be the set of all functions from $\Z$ to $\R\text{.}$ Prove that pointwise multiplication on $G$ (that is, the operation defined by $(fg)(x)=f(x)g(x)$ for all $f,g\in G$ and $x\in \Z$) is commutative. (Note. To prove that two functions, $h$ and $j\text{,}$ sharing the same domain $D$ are equal, you need to show that $h(x)=j(x)$ for every $x\in D\text{.}$)

Solution
##### Exercise3

Decide which of the following binary structures are groups. For each, if the binary structure isn't a group, prove that. (Remember, you should not state that inverses do or do not exist for elements until you have made sure that the structure contains an identity element!) If the binary structure is a group, prove that.

1. $\Q$ under multiplication

2. $\M_2(\R)$ under addition

3. $\M_2(\R)$ under multiplication

4. $\R^+$ under $*\text{,}$ defined by $a*b=\sqrt{ab}$ for all $a,b\in \R^+$

Solution
##### Exercise4

Give an example of an abelian group containing 711 elements.

Solution
##### Exercise5

Let $n\in \Z\text{.}$ Prove that $n\Z$ is a group under the usual addition of integers. Note: You may use the fact that $\langle n\Z,+\rangle$ is a binary structure if you provide a reference for this fact.

Solution
##### Exercise6

Let $n\in \Z^+\text{.}$ Prove that $SL(n,\R)$ is a group under matrix multiplication. Note: You may use the fact that $\langle SL(n\R),\cdot\rangle$ is a binary structure if you provide a reference for this fact.

Solution
##### Exercise7

1. List three distinct integers that are congruent to $6$ modulo $5\text{.}$

2. List the elements of $\Z_5\text{.}$

3. Compute:

1. $4+5$ in $\Z\text{;}$

2. $4+5$ in $\Q\text{;}$

3. $4+_65$ in $\Z_6\text{;}$

4. the inverse of $4$ in $\Z\text{;}$

5. the inverse of $4$ in $\Z_6\text{.}$

4. Why does it not make sense for me to ask you to compute $4+_3 2$ in $\Z_3\text{?}$ Please answer this using a complete, grammatically correct sentence.

Solution
##### Exercise8

Let $G$ be a group with identity element $e\text{.}$ Prove that if every element of $G$ is its own inverse, then $G$ is abelian.

Solution
##### Exercise9

Let $G$ be a group. The subset

\begin{equation*} Z(G):=\{z \in G\,:\, zg=gz \mbox{ for all } g\in G\} \end{equation*}

of $G$ is called the center of $G\text{.}$ In other words, $Z(G)$ is the set of all elements of $G$ that commute with every element of $G\text{.}$ Prove that $Z(G)$ is closed in $G\text{.}$

Solution

# Exercises3.4Exercises

##### Exercise1

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let $G$ and $G'$ be groups.

1. If there exists a homomorphism $\phi\,:\,G\to G'\text{,}$ then $G$ and $G'$ must be isomorphic groups.

2. There is an integer $n\geq 2$ such that $\Z\simeq \Z_n\text{.}$

3. If $|G|=|G'|=3\text{,}$ then we must have $G\simeq G'\text{.}$

4. If $|G|=|G'|=4\text{,}$ then we must have $G\simeq G'\text{.}$

Solution
##### Exercise2

For each of the following functions, prove or disprove that the function is (i) a homomorphism; (ii) an isomorphism. (Remember to work with the default operation on each of these groups!)

1. The function $f:\Z\to\Z$ defined by $f(n)=2n\text{.}$

2. The function $g:\R\to\R$ defined by $g(x)=x^2\text{.}$

3. The function $h:\Q^*\to\Q^*$ defined by $h(x)=x^2\text{.}$

Solution
##### Exercise3

LDefine $d : GL(2,\R)\to \R^*$ by $d(A)=\det A\text{.}$ Prove/disprove that $d$ is:

1. a homomorphism

2. 1-1

3. onto

4. an isomorphism.

Solution
##### Exercise4

Complete the group tables for $\Z_4$ and $\Z_8^{\times}\text{.}$ Use the group tables to decide whether or not $\Z_4$ and $\Z_8^{\times}$ are isomorphic to one another. (You do not need to provide a proof.)

Solution
##### Exercise5

Let $n\in \Z^+\text{.}$ Prove that $\langle n\Z,+\rangle \simeq \langle \Z,+\rangle\text{.}$

Solution
##### Exercise6

1. Let $G$ and $G'$ be groups, where $G$ is abelian and $G\simeq G'\text{.}$ Prove that $G'$ is abelian.

2. Give an example of groups $G$ and $G'\text{,}$ where $G$ is abelian and there exists a homomorphism from $G$ to $G'\text{,}$ but $G'$ is NOT abelian.

Solution
##### Exercise7

Let $\langle G,\cdot\rangle$ and $\langle G',\cdot'\rangle$ be groups with identity elements $e$ and $e'\text{,}$ respectively, and let $\phi$ be a homomorphism from $G$ to $G'\text{.}$ Let $a\in G\text{.}$ Prove that $\phi(a)^{-1}=\phi(a^{-1})\text{.}$

Solution

# Exercises4.3Exercises

##### Exercise1

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let $G$ and $G'$ be groups.

1. Every group contains at least two distinct subgroups.

2. If $H$ is a proper subgroup of group $G$ and $G$ is finite, then we must have $|H|\lt |G|\text{.}$

3. $7\Z$ is a subgroup of $14\Z\text{.}$

4. A group $G$ may have two distinct proper subgroups which are isomorphic (to one another).

Solution
##### Exercise2

Give specific, precise examples of the following groups $G$ with subgroups $H\text{:}$

1. A group $G$ with a proper subgroup $H$ of $G$ such that $|H|=|G|\text{.}$

2. A group $G$ of order 12 containing a subgroup $H$ with $|H|=3\text{.}$

3. A nonabelian group $G$ containing a nontrivial abelian subgroup $H\text{.}$

4. A finite subgroup $H$ of an infinite group $G\text{.}$

Solution
##### Exercise3

Let $n\in \Z^+\text{.}$

1. Prove that $n\Z \leq \Z\text{.}$

2. Prove that the set $H=\{A\in \M_n(\R)\,:\,\det A=\pm 1\}$ is a subgroup of $GL(n,\R)\text{.}$

(Note: Your proofs do not need to be long to be correct!)

Solution
##### Exercise4

Let $n\in \Z^+\text{.}$ For each group $G$ and subset $H\text{,}$ decide whether or not $H$ is a subgroup of $G\text{.}$ In the cases in which $H$ is not a subgroup of $G\text{,}$ provide a proof. (Note. Your proofs do not need to be long to be correct!)

1. $G=\R\text{,}$ $H=\Z$

2. $G=\Z_{15}\text{,}$ $H=\{0,5,10\}$

3. $G=\Z_{15}\text{,}$ $H=\{0,4,8,12\}$

4. $G=\C\text{,}$ $H=\R^*$

5. $G=\C^*\text{,}$ $H=\{1,i,-1,-i\}$

6. $G=\M_n(\R)\text{,}$ $H=GL(n,\R)$

7. $G=GL(n,\R)\text{,}$ $H=\{A\in \M_n(\R)\,:\,\det A = -1\}$

Solution
##### Exercise5

Let $G$ and $G'$ be groups, let $\phi$ be a homomorphism from $G$ to $G'\text{,}$ and let $H$ be a subgroup of $G\text{.}$ Prove that $\phi(H)$ is a subgroup of $G'\text{.}$

Solution
##### Exercise6

Let $G$ be an abelian group, and let $U=\{g\in G\,:\, g^{-1}=g\}.$ Prove that $U$ is a subgroup of $G\text{.}$

Solution

# Exercises5.3Exercises

##### Exercise1

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let $G$ be a group with identity element $e\text{.}$

1. If $G$ is infinite and cyclic, then $G$ must have infinitely many generators.

2. There may be two distinct elements $a$ and $b$ of a group $G$ with $\langle a\rangle =\langle b\rangle\text{.}$

3. If $a,b\in G$ and $a\in \langle b\rangle$ then we must have $b\in \langle a\rangle\text{.}$

4. If $a\in G$ with $a^4=e\text{,}$ then $o(a)$ must equal $4\text{.}$

5. If $G$ is countable then $G$ must be cyclic.

Solution
##### Exercise2

Give examples of the following.

1. An infinite noncyclic group $G$ containing an infinite cyclic subgroup $H\text{.}$

2. An infinite noncyclic group $G$ containing a finite nontrivial cyclic subgroup $H\text{.}$

3. A cyclic group $G$ containing exactly 20 elements.

4. A nontrivial cyclic group $G$ whose elements are all matrices.

5. A noncyclic group $G$ such that every proper subgroup of $G$ is cyclic.

Solution
##### Exercise3

Find the orders of the following elements in the given groups.

1. $2\in \Z$

2. $-i\in \C^*$

3. $-I_2\in GL(2,\R)$

4. $-I_2\in \M_2(\R)$

5. $(6,8)\in \Z_{10}\times \Z_{10}$

Solution
##### Exercise4

For each of the following, if the group is cyclic, list all of its generators. If the group is not cyclic, write NC.

1. $5\Z$

2. $\Z_{18}$

3. $\R$

4. $\langle \pi\rangle$ in $\R$

5. $\Z_2^2$

6. $\langle 8\rangle$ in $\Q^*$

Solution
##### Exercise5

Explicitly identify the elements of the following subgroups of the given groups. You may use set-builder notation if the subgroup is infinite, or a conventional name for the subgroup if we have one.

1. $\langle 3\rangle$ in $\Z$

2. $\langle i\rangle$ in $C^*$

3. $\langle A\rangle\text{,}$ for $A=\left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 0 \end{array} \right]\in \M_2(\R)$

4. $\langle (2,3)\rangle$ in $\Z_4\times \Z_5$

5. $\langle B\rangle\text{,}$ for $B=\left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\in GL(2,\R)$

Solution
##### Exercise6

Draw subgroup lattices for the following groups.

1. $\Z_6$

2. $\Z_{13}$

3. $\Z_{18}$

Solution
##### Exercise7

Let $G$ be a group with no nontrivial proper subgroups. Prove that $G$ is cyclic.

Solution

# Exercises6.6Exercises

##### Exercise1

Let $\sigma=(134)\text{,}$ $\tau=(23)(145)\text{,}$ $\rho=(56)(78)\text{,}$ and $\alpha=(12)(145)$ in $S_8\text{.}$ Compute the following.

1. $\sigma \tau$

2. $\tau \sigma$

3. $\tau^2$

4. $\tau^{-1}$

5. $o(\tau)$

6. $o(\rho)$

7. $o(\alpha)$

8. $\langle \tau\rangle$

Solution
##### Exercise2

Prove Lemma 6.3.4.

Solution
##### Exercise3

Prove that $A_n$ is a subgroup of $S_n\text{.}$

Solution
##### Exercise4

Prove or disprove: The set of all odd permutations in $S_n$ is a subgroup of $S_n\text{.}$

Solution
##### Exercise5

Let $n$ be an integer greater than 2. $m \in \{1,2,\ldots,n\}\text{,}$ and let $H=\{\sigma\in S_n\,:\,\sigma(m)=m\}$ (in other words, $H$ is the set of all permutations in $S_n$ that fix $m$).

1. Prove that $H\leq S_n\text{.}$

2. Identify a familiar group to which $H$ is isomorphic. (You do not need to show any work.)

Solution
##### Exercise6

Write $rfr^2frfr$ in $D_5$ in standard form.

Solution
##### Exercise7

Prove or disprove: $D_6\simeq S_6\text{.}$

Solution
##### Exercise8

Which elements of $D_4$ (if any)

1. have order 2?

2. have order $3\text{?}$

Solution
##### Exercise9

Let $n$ be an even integer that's greater than or equal to 4. Prove that $r^{n/2}\in Z(D_n)\text{:}$ that is, prove that $r^{n/2}$ commutes with every element of $D_n\text{.}$ (Do NOT simply refer to the last statement in Theorem 6.5.10; that is the statement you are proving here.)

Solution

# Exercises7.4Exercises

##### Exercise1

How many distinct partitions of the set $S=\{a,b,c,d\}$ are there? You do not need to list them. (Yes, you can find this answer online. But I recommend doing the work yourself for practice working with partitions!)

Solution
##### Exercise2

1. Let $n\in \Z^+\text{.}$ Prove that $\equiv_n$ is an equivalence relation on $\Z\text{.}$

2. The cells of the induced partition of $\Z$ are called the residue classes (or congruence classes) of $\Z$ modulo $n$. Using set notation of the form $\{\ldots,\#, \#,\#,\ldots\}$ for each class, write down the residue classes of $\Z$ modulo $4\text{.}$

Solution
##### Exercise3

Let $G$ be a group with subgroup $H\text{.}$ Prove that $\simr$ is an equivalence relation on $G\text{.}$

Solution
##### Exercise4

Find the indices of:

1. $H=\langle (15)(24)\rangle$ in $S_5$

2. $K=\langle (2354)(34)\rangle$ in $S_6$

3. $A_n$ in $S_n$

Solution
##### Exercise5

For each subgroup $H$ of group $G\text{,}$ (i) find the left and the right cosets of $H$ in $G\text{,}$ (ii) decide whether or not $H$ is normal in $G\text{,}$ and (iii) find $(G:H)\text{.}$

Write all permutations using disjoint cycle notation, and write all dihedral group elements using standard form.

1. $H=6\Z$ in $G=2\Z$

2. $H=\langle 4\rangle$ in $\Z_{20}$

3. $H=\langle (23)\rangle$ in $G=S_3$

4. $H=\langle r\rangle$ in $G=D_4$

5. $H=\langle f\rangle$ in $G=D_4$

Solution
##### Exercise6

For each of the following, give an example of a group $G$ with a subgroup $H$ that matches the given conditions. If no such example exists, prove that.

1. A group $G$ with subgroup $H$ such that $|G/H|=1\text{.}$

2. A finite group $G$ with subgroup $H$ such that $|G/H|=|G|\text{.}$

3. An abelian group $G$ of order $8$ containing a non-normal subgroup $H$ of order 2.

4. A group $G$ of order 8 containing a normal subgroup of order $2\text{.}$

5. A nonabelian group $G$ of order 8 containing a normal subgroup of index $2\text{.}$

6. A group $G$ of order 8 containing a subgroup of order $3\text{.}$

7. An infinite group $G$ containing a subgroup $H$ of finite index.

8. An infinite group $G$ containing a finite nontrivial subgroup $H\text{.}$

Solution
##### Exercise7

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let $G$ be a group with subgroup $H$ and elements $a,b\in G\text{.}$

1. If $a\in bH$ then $aH$ must equal $bH\text{.}$

2. $aH$ must equal $Ha\text{.}$

3. If $aH=bH$ then $Ha$ must equal $Hb\text{.}$

4. If $a\in H$ then $aH$ must equal $Ha\text{.}$

5. $H$ must be normal in $G$ if there exists $a\in G$ such that $aH=Ha\text{.}$

6. If $aH=bH$ then $ah=bh$ for every $h\in H\text{.}$

7. $|G/H|$ must be less than $|G|\text{.}$

8. $(G:H)$ must be less than or equal to $|G|\text{.}$

Solution
##### Exercise8

Let $G$ be a group of order $pq\text{,}$ where $p$ and $q$ are prime, and let $H$ be a proper subgroup of $G\text{.}$ Prove that $H$ is cyclic.

Solution
##### Exercise9

Prove Corollary 7.3.10: that is, let $G$ be a group of prime order, and prove that $G$ is cyclic.

Solution
##### Exercise10

Let $G$ be a group of finite order $n\text{,}$ containing identity element $e\text{.}$ Let $a\in G\text{.}$ Prove that $a^n=e\text{.}$

Solution

# Exercises8.4Exercises

##### Exercise1

Let $G$ be a group and let $H\leq G$ have index 2. Prove that $H\unlhd G\text{.}$

Solution
##### Exercise2

Let $G$ be an abelian group with $N\unlhd G\text{.}$ Prove that $G/N$ is abelian.

Solution
##### Exercise3

Find the following.

1. $|2\Z/6\Z|$
2. $|H|\text{,}$ for $H=2+\langle 6\rangle \subseteq \Z_{12}$
3. $o(2+\langle 6\rangle)$ in $\Z_{12}/\langle 6\rangle$
4. $\langle f+H\rangle$ in $D_4/H\text{,}$ where $H=\{e,r^2\}$
5. $|(\Z_6\times \Z_8)/(\langle 3\rangle\times \langle 2\rangle)|$
6. $|(\Z_{15} \times \Z_{24})/\langle (5,4)\rangle|$
Solution
##### Exercise4

For each of the following, find a familiar group to which the given group is isomorphic. (Hint: Consider the group order, properties such as abelianness and cyclicity, group tables, orders of elements, etc.)

1. $\Z/14\Z$
2. $3\Z/12\Z$
3. $S_8/A_8$
4. $(4\Z \times 15\Z)/(\langle 2 \rangle \times \langle 3 \rangle )$
5. $D_4/\langle r^2 \rangle$
Solution
##### Exercise5

Let $H\unlhd G$ with index $k\text{,}$ and let $a\in G\text{.}$ Prove that $a^k\in H\text{.}$

Solution

# Exercises9.3Exercises

##### Exercise1

Let $F$ be the group of all functions from $[0,1]$ to $\R\text{,}$ under pointwise addition. Let

\begin{equation*} N=\{f\in F: f(1/4)=0\}. \end{equation*}

Prove that $F/N$ is a group that's isomorphic to $\R\text{.}$

Solution
##### Exercise2

Let $N=\{1,-1\}\subseteq \R^*\text{.}$ Prove that $\R^*/N$ is a group that's isomorphic to $\R^+\text{.}$

Solution
##### Exercise3

Let $n\in \Z^+$ and let $H=\{A\in GL(n,\R)\,:\, \det A =\pm 1\}\text{.}$ Identify a group familiar to us that is isomorphic to $GL(n,\R)/H\text{.}$

Solution
##### Exercise4

Let $G$ and $G'$ be groups with respective normal subgroups $N$ and $N'\text{.}$ Prove or disprove: If $G/N\simeq G'/N'$ then $G\simeq G'\text{.}$

Solution