###### 1.

Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by $D_4\text{.}$

###### 2.

Give an example of two elements $A$ and $B$ in $GL_2({\mathbb R})$ with $AB \neq BA\text{.}$

Hint

Pick two matrices. Almost any pair will work.

###### 3.

Prove that the product of two matrices in $SL_2({\mathbb R})$ has determinant one.

###### 4.

Prove that the set of matrices of the form

\begin{equation*} \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} \end{equation*}

is a group under matrix multiplication. This group, known as the Heisenberg group, is important in quantum physics. Matrix multiplication in the Heisenberg group is defined by

\begin{equation*} \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & x' & y' \\ 0 & 1 & z' \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & x+x' & y+y'+xz' \\ 0 & 1 & z+z' \\ 0 & 0 & 1 \end{pmatrix}\text{.} \end{equation*}
###### 5.

Prove that $\det(AB) = \det(A) \det(B)$ in $GL_2({\mathbb R})\text{.}$ Use this result to show that the binary operation in the group $GL_2({\mathbb R})$ is closed; that is, if $A$ and $B$ are in $GL_2({\mathbb R})\text{,}$ then $AB \in GL_2({\mathbb R})\text{.}$

###### 6.

Let ${\mathbb Z}_2^n = \{ (a_1, a_2, \ldots, a_n) : a_i \in {\mathbb Z}_2 \}\text{.}$ Define a binary operation on ${\mathbb Z}_2^n$ by

\begin{equation*} (a_1, a_2, \ldots, a_n) + (b_1, b_2, \ldots, b_n) = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n)\text{.} \end{equation*}

Prove that ${\mathbb Z}_2^n$ is a group under this operation. This group is important in algebraic coding theory.

###### 7.

Show that ${\mathbb R}^{\ast} = {\mathbb R} \setminus \{0 \}$ is a group under the operation of multiplication.

###### 8.

Given the groups ${\mathbb R}^{\ast}$ and ${\mathbb Z}\text{,}$ let $G = {\mathbb R}^{\ast} \times {\mathbb Z}\text{.}$ Define a binary operation $\circ$ on $G$ by $(a,m) \circ (b,n) = (ab, m + n)\text{.}$ Show that $G$ is a group under this operation.

###### 9.

Prove or disprove that every group containing six elements is abelian.

Hint

There is a nonabelian group containing six elements.

###### 10.

Give a specific example of some group $G$ and elements $g, h \in G$ where $(gh)^n \neq g^nh^n\text{.}$

Hint

Look at the symmetry group of an equilateral triangle or a square.

###### 11.

Give an example of three different groups with eight elements. Why are the groups different?

Hint

The are five different groups of order 8.

###### 12.

Show that there are $n!$ permutations of a set containing $n$ items.

Hint

Let

\begin{equation*} \sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{pmatrix} \end{equation*}

be in $S_n\text{.}$ All of the $a_i$s must be distinct. There are $n$ ways to choose $a_1\text{,}$ $n - 1$ ways to choose $a_2, \ldots\text{,}$ 2 ways to choose $a_{n - 1}\text{,}$ and only one way to choose $a_n\text{.}$ Therefore, we can form $\sigma$ in $n(n - 1) \cdots 2 \cdot 1 = n!$ ways.

###### 13.

Show that

\begin{equation*} 0 + a \equiv a + 0 \equiv a \pmod{ n } \end{equation*}

for all $a \in {\mathbb Z}_n\text{.}$

###### 14.

Prove that there is a multiplicative identity for the integers modulo $n\text{:}$

\begin{equation*} a \cdot 1 \equiv a \pmod{n}\text{.} \end{equation*}
###### 15.

For each $a \in {\mathbb Z}_n$ find an element $b \in {\mathbb Z}_n$ such that

\begin{equation*} a + b \equiv b + a \equiv 0 \pmod{ n}\text{.} \end{equation*}
###### 16.

Show that addition and multiplication mod $n$ are well defined operations. That is, show that the operations do not depend on the choice of the representative from the equivalence classes mod $n\text{.}$

###### 17.

Show that addition and multiplication mod $n$ are associative operations.

###### 18.

Show that multiplication distributes over addition modulo $n\text{:}$

\begin{equation*} a(b + c) \equiv ab + ac \pmod{n}\text{.} \end{equation*}
###### 19.

Let $a$ and $b$ be elements in a group $G\text{.}$ Prove that $ab^na^{-1} = (aba^{-1})^n$ for $n \in \mathbb Z\text{.}$

Hint
\begin{align*} (aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})\\ & = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}\\ & = ab^na^{-1}\text{.} \end{align*}
###### 20.

Let $U(n)$ be the group of units in ${\mathbb Z}_n\text{.}$ If $n \gt 2\text{,}$ prove that there is an element $k \in U(n)$ such that $k^2 = 1$ and $k \neq 1\text{.}$

###### 21.

Prove that the inverse of $g _1 g_2 \cdots g_n$ is $g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}\text{.}$

###### 22.

Prove the remainder of Proposition 2.21 : if $G$ is a group and $a, b \in G\text{,}$ then the equation $xa = b$ has a unique solution in $G\text{.}$

###### 24.

Prove the right and left cancellation laws for a group $G\text{;}$ that is, show that in the group $G\text{,}$ $ba = ca$ implies $b = c$ and $ab = ac$ implies $b = c$ for elements $a, b, c \in G\text{.}$

###### 25.

Show that if $a^2 = e$ for all elements $a$ in a group $G\text{,}$ then $G$ must be abelian.

Hint

Since $abab = (ab)^2 = e = a^2 b^2 = aabb\text{,}$ we know that $ba = ab\text{.}$

###### 26.

Show that if $G$ is a finite group of even order, then there is an $a \in G$ such that $a$ is not the identity and $a^2 = e\text{.}$

###### 27.

Let $G$ be a group and suppose that $(ab)^2 = a^2b^2$ for all $a$ and $b$ in $G\text{.}$ Prove that $G$ is an abelian group.

###### 28.

Find all the subgroups of ${\mathbb Z}_3 \times {\mathbb Z}_3\text{.}$ Use this information to show that ${\mathbb Z}_3 \times {\mathbb Z}_3$ is not the same group as ${\mathbb Z}_9\text{.}$ (See Example 2.28 for a short description of the product of groups.)

###### 29.

Find all the subgroups of the symmetry group of an equilateral triangle.

Hint

$H_1 = \{ \identity \}\text{,}$ $H_2 = \{ \identity, \rho_1, \rho_2 \}\text{,}$ $H_3 = \{ \identity, \mu_1 \}\text{,}$ $H_4 = \{ \identity, \mu_2 \}\text{,}$ $H_5 = \{ \identity, \mu_3 \}\text{,}$ $S_3\text{.}$

###### 30.

Compute the subgroups of the symmetry group of a square.

###### 31.

Let $H = \{2^k : k \in {\mathbb Z} \}\text{.}$ Show that $H$ is a subgroup of ${\mathbb Q}^*\text{.}$

###### 32.

Let $n = 0, 1, 2, \ldots$ and $n {\mathbb Z} = \{ nk : k \in {\mathbb Z} \}\text{.}$ Prove that $n {\mathbb Z}$ is a subgroup of ${\mathbb Z}\text{.}$ Show that these subgroups are the only subgroups of $\mathbb{Z}\text{.}$

###### 33.

Let ${\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}\text{.}$ Prove that ${\mathbb T}$ is a subgroup of ${\mathbb C}^*\text{.}$

###### 34.

Let $G$ consist of the $2 \times 2$ matrices of the form

\begin{equation*} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\text{,} \end{equation*}

where $\theta \in {\mathbb R}\text{.}$ Prove that $G$ is a subgroup of $SL_2({\mathbb R})\text{.}$

###### 35.

Prove that

\begin{equation*} G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \} \end{equation*}

is a subgroup of ${\mathbb R}^{\ast}$ under the group operation of multiplication.

Hint

The identity of $G$ is $1 = 1 + 0 \sqrt{2}\text{.}$ Since $(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}\text{,}$ $G$ is closed under multiplication. Finally, $(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}$

###### 36.

Let $G$ be the group of $2 \times 2$ matrices under addition and

\begin{equation*} H = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a + d = 0 \right\}\text{.} \end{equation*}

Prove that $H$ is a subgroup of $G\text{.}$

###### 37.

Prove or disprove: $SL_2( {\mathbb Z} )\text{,}$ the set of $2 \times 2$ matrices with integer entries and determinant one, is a subgroup of $SL_2( {\mathbb R} )\text{.}$

###### 38.

List the subgroups of the quaternion group, $Q_8\text{.}$

###### 39.

Prove that the intersection of two subgroups of a group $G$ is also a subgroup of $G\text{.}$

###### 40.

Prove or disprove: If $H$ and $K$ are subgroups of a group $G\text{,}$ then $H \cup K$ is a subgroup of $G\text{.}$

Hint

Look at $S_3\text{.}$

###### 41.

Prove or disprove: If $H$ and $K$ are subgroups of a group $G\text{,}$ then $H K = \{hk : h \in H \text{ and } k \in K \}$ is a subgroup of $G\text{.}$ What if $G$ is abelian?

###### 42.

Let $G$ be a group and $g \in G\text{.}$ Show that

\begin{equation*} Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \} \end{equation*}

is a subgroup of $G\text{.}$ This subgroup is called the center of $G\text{.}$

###### 43.

Let $a$ and $b$ be elements of a group $G\text{.}$ If $a^4 b = ba$ and $a^3 = e\text{,}$ prove that $ab = ba\text{.}$

Hint

$b a = a^4 b = a^3 a b = ab$

###### 44.

Give an example of an infinite group in which every nontrivial subgroup is infinite.

###### 45.

If $xy = x^{-1} y^{-1}$ for all $x$ and $y$ in $G\text{,}$ prove that $G$ must be abelian.

###### 46.

Prove or disprove: Every proper subgroup of a nonabelian group is nonabelian.

###### 47.

Let $H$ be a subgroup of $G$ and

\begin{equation*} C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}\text{.} \end{equation*}

Prove $C(H)$ is a subgroup of $G\text{.}$ This subgroup is called the centralizer of $H$ in $G\text{.}$

###### 48.

Let $H$ be a subgroup of $G\text{.}$ If $g \in G\text{,}$ show that $gHg^{-1} = \{ghg^{-1} : h\in H\}$ is also a subgroup of $G\text{.}$

###### 49.

Prove or disprove each of the following statements.

1. All of the generators of ${\mathbb Z}_{60}$ are prime.

2. $U(8)$ is cyclic.

3. ${\mathbb Q}$ is cyclic.

4. If every proper subgroup of a group $G$ is cyclic, then $G$ is a cyclic group.

5. A group with a finite number of subgroups is finite.

Hint

(a) False; (c) false; (e) true.

###### 50.

Find the order of each of the following elements.

1. $\displaystyle 5 \in {\mathbb Z}_{12}$

2. $\displaystyle \sqrt{3} \in {\mathbb R}$

3. $\displaystyle \sqrt{3} \in {\mathbb R}^\ast$

4. $\displaystyle -i \in {\mathbb C}^\ast$

5. $\displaystyle 72 \in {\mathbb Z}_{240}$

6. $\displaystyle 312 \in {\mathbb Z}_{471}$

Hint

(a) $12\text{;}$ (c) infinite; (e) $10\text{.}$

###### 51.

List all of the elements in each of the following subgroups.

1. The subgroup of ${\mathbb Z}$ generated by $7$

2. The subgroup of ${\mathbb Z}_{24}$ generated by $15$

3. All subgroups of ${\mathbb Z}_{12}$

4. All subgroups of ${\mathbb Z}_{60}$

5. All subgroups of ${\mathbb Z}_{13}$

6. All subgroups of ${\mathbb Z}_{48}$

7. The subgroup generated by 3 in $U(20)$

8. The subgroup generated by 5 in $U(18)$

9. The subgroup of ${\mathbb R}^\ast$ generated by $7$

10. The subgroup of ${\mathbb C}^\ast$ generated by $i$ where $i^2 = -1$

11. The subgroup of ${\mathbb C}^\ast$ generated by $2i$

12. The subgroup of ${\mathbb C}^\ast$ generated by $(1 + i) / \sqrt{2}$

13. The subgroup of ${\mathbb C}^\ast$ generated by $(1 + \sqrt{3}\, i) / 2$

Hint

(a) $7 {\mathbb Z} = \{ \ldots, -7, 0, 7, 14, \ldots \}\text{;}$ (b) $\{ 0, 3, 6, 9, 12, 15, 18, 21 \}\text{;}$ (c) $\{ 0 \}\text{,}$ $\{ 0, 6 \}\text{,}$ $\{ 0, 4, 8 \}\text{,}$ $\{ 0, 3, 6, 9 \}\text{,}$ $\{ 0, 2, 4, 6, 8, 10 \}\text{;}$ (g) $\{ 1, 3, 7, 9 \}\text{;}$ (j) $\{ 1, -1, i, -i \}\text{.}$

###### 52.

Find the subgroups of $GL_2( {\mathbb R })$ generated by each of the following matrices.

1. $\displaystyle \displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

2. $\displaystyle \displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}$

3. $\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}$

4. $\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$

5. $\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}$

6. $\displaystyle \displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}$

Hint

(a)

\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\text{.} \end{equation*}

(c)

\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} -1 & 1 \\ -1 & 0 \end{pmatrix}, \\ \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\text{.} \end{equation*}
###### 53.

Find the order of every element in ${\mathbb Z}_{18}\text{.}$

###### 54.

Find the order of every element in the symmetry group of the square, $D_4\text{.}$

###### 55.

What are all of the cyclic subgroups of the quaternion group, $Q_8\text{?}$

###### 56.

List all of the cyclic subgroups of $U(30)\text{.}$

###### 57.

List every generator of each subgroup of order 8 in ${\mathbb Z}_{32}\text{.}$

###### 58.

Find all elements of finite order in each of the following groups. Here the “$\ast$” indicates the set with zero removed.

1. $\displaystyle {\mathbb Z}$

2. $\displaystyle {\mathbb Q}^\ast$

3. $\displaystyle {\mathbb R}^\ast$

Hint

(a) $0\text{;}$ (b) $1, -1\text{.}$

###### 59.

If $a^{24} =e$ in a group $G\text{,}$ what are the possible orders of $a\text{?}$

Hint

$1, 2, 3, 4, 6, 8, 12, 24\text{.}$

###### 60.

Find a cyclic group with exactly one generator. Can you find cyclic groups with exactly two generators? Four generators? How about $n$ generators?

###### 61.

For $n \leq 20\text{,}$ which groups $U(n)$ are cyclic? Make a conjecture as to what is true in general. Can you prove your conjecture?

###### 62.

Let

\begin{equation*} A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \qquad \text{and} \qquad B = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix} \end{equation*}

be elements in $GL_2( {\mathbb R} )\text{.}$ Show that $A$ and $B$ have finite orders but $AB$ does not.

###### 63.

Prove each of the following statements.

1. $\displaystyle |z| = | \overline{z}|$

2. $\displaystyle z \overline{z} = |z|^2$

3. $\displaystyle z^{-1} = \overline{z} / |z|^2$

4. $\displaystyle |z +w| \leq |z| + |w|$

5. $\displaystyle |z - w| \geq | |z| - |w||$

6. $\displaystyle |z w| = |z| |w|$

###### 64.

List and graph the 6th roots of unity. What are the generators of this group? What are the primitive 6th roots of unity?

###### 65.

List and graph the 5th roots of unity. What are the generators of this group? What are the primitive 5th roots of unity?

###### 66.

Calculate each of the following.

1. $\displaystyle 292^{3171} \pmod{ 582}$

2. $\displaystyle 2557^{ 341} \pmod{ 5681}$

3. $\displaystyle 2071^{ 9521} \pmod{ 4724}$

4. $\displaystyle 971^{ 321} \pmod{ 765}$

Hint

(a) $292\text{;}$ (c) $1523\text{.}$

###### 67.

Let $a, b \in G\text{.}$ Prove the following statements.

1. The order of $a$ is the same as the order of $a^{-1}\text{.}$

2. For all $g \in G\text{,}$ $|a| = |g^{-1}ag|\text{.}$

3. The order of $ab$ is the same as the order of $ba\text{.}$

###### 68.

Let $p$ and $q$ be distinct primes. How many generators does ${\mathbb Z}_{pq}$ have?

###### 69.

Let $p$ be prime and $r$ be a positive integer. How many generators does ${\mathbb Z}_{p^r}$ have?

###### 70.

Prove that ${\mathbb Z}_{p}$ has no nontrivial subgroups if $p$ is prime.

###### 71.

If $g$ and $h$ have orders $15$ and $16$ respectively in a group $G\text{,}$ what is the order of $\langle g \rangle \cap \langle h \rangle \text{?}$

Hint

$|\langle g \rangle \cap \langle h \rangle| = 1\text{.}$

###### 72.

Let $a$ be an element in a group $G\text{.}$ What is a generator for the subgroup $\langle a^m \rangle \cap \langle a^n \rangle\text{?}$

###### 73.

Prove that ${\mathbb Z}_n$ has an even number of generators for $n \gt 2\text{.}$

###### 74.

Suppose that $G$ is a group and let $a\text{,}$ $b \in G\text{.}$ Prove that if $|a| = m$ and $|b| = n$ with $\gcd(m,n) = 1\text{,}$ then $\langle a \rangle \cap \langle b \rangle = \{ e \}\text{.}$

###### 75.

Let $G$ be an abelian group. Show that the elements of finite order in $G$ form a subgroup. This subgroup is called the torsion subgroup of $G\text{.}$

Hint

The identity element in any group has finite order. Let $g, h \in G$ have orders $m$ and $n\text{,}$ respectively. Since $(g^{-1})^m = e$ and $(gh)^{mn} = e\text{,}$ the elements of finite order in $G$ form a subgroup of $G\text{.}$

###### 76.

Let $G$ be a finite cyclic group of order $n$ generated by $x\text{.}$ Show that if $y = x^k$ where $\gcd(k,n) = 1\text{,}$ then $y$ must be a generator of $G\text{.}$

###### 77.

If $G$ is an abelian group that contains a pair of cyclic subgroups of order $2\text{,}$ show that $G$ must contain a subgroup of order $4\text{.}$ Does this subgroup have to be cyclic?

###### 78.

Let $G$ be an abelian group of order $pq$ where $\gcd(p,q) = 1\text{.}$ If $G$ contains elements $a$ and $b$ of order $p$ and $q$ respectively, then show that $G$ is cyclic.

###### 79.

Prove that the subgroups of $\mathbb Z$ are exactly $n{\mathbb Z}$ for $n = 0, 1, 2, \ldots\text{.}$

###### 80.

Prove that the generators of ${\mathbb Z}_n$ are the integers $r$ such that $1 \leq r \lt n$ and $\gcd(r,n) = 1\text{.}$

###### 81.

Prove that if $G$ has no proper nontrivial subgroups, then $G$ is a cyclic group.

Hint

If $g$ is an element distinct from the identity in $G\text{,}$ $g$ must generate $G\text{;}$ otherwise, $\langle g \rangle$ is a nontrivial proper subgroup of $G\text{.}$

###### 82.

Prove that the order of an element in a cyclic group $G$ must divide the order of the group.

###### 83.

Prove that if $G$ is a cyclic group of order $m$ and $d \mid m\text{,}$ then $G$ must have a subgroup of order $d\text{.}$

###### 84.

For what integers $n$ is $-1$ an $n$th root of unity?

###### 85.

If $z = r( \cos \theta + i \sin \theta)$ and $w = s(\cos \phi + i \sin \phi)$ are two nonzero complex numbers, show that

\begin{equation*} zw = rs[ \cos( \theta + \phi) + i \sin( \theta + \phi)]\text{.} \end{equation*}
###### 86.

Prove that the circle group is a subgroup of ${\mathbb C}^*\text{.}$

###### 87.

Prove that the $n$th roots of unity form a cyclic subgroup of ${\mathbb T}$ of order $n\text{.}$

###### 88.

Let $\alpha \in \mathbb T\text{.}$ Prove that $\alpha^m =1$ and $\alpha^n = 1$ if and only if $\alpha^d = 1$ for $d = \gcd(m,n)\text{.}$

###### 89.

Let $z \in {\mathbb C}^\ast\text{.}$ If $|z| \neq 1\text{,}$ prove that the order of $z$ is infinite.

###### 90.

Let $z =\cos \theta + i \sin \theta$ be in ${\mathbb T}$ where $\theta \in {\mathbb Q}\text{.}$ Prove that the order of $z$ is infinite.

###### 91.

Find $(a_1, a_2, \ldots, a_n)^{-1}\text{.}$

Hint

$(a_1, a_2, \ldots, a_n)^{-1} = (a_1, a_{n}, a_{n-1}, \ldots, a_2)$

###### 92.

What are the possible cycle structures of elements of $A_5\text{?}$ What about $A_6\text{?}$

Hint

Permutations of the form

\begin{equation*} (1), (a_1, a_2)(a_3, a_4), (a_1, a_2, a_3), (a_1, a_2, a_3, a_4, a_5) \end{equation*}

are possible for $A_5\text{.}$

###### 93.

Let $\sigma \in S_n$ have order $n\text{.}$ Show that for all integers $i$ and $j\text{,}$ $\sigma^i = \sigma^j$ if and only if $i \equiv j \pmod{n}\text{.}$

###### 94.

Let $\sigma = \sigma_1 \cdots \sigma_m \in S_n$ be the product of disjoint cycles. Prove that the order of $\sigma$ is the least common multiple of the lengths of the cycles $\sigma_1, \ldots, \sigma_m\text{.}$

###### 95.

Using cycle notation, list the elements in $D_5\text{.}$ What are $r$ and $s\text{?}$ Write every element as a product of $r$ and $s\text{.}$

###### 96.

If the diagonals of a cube are labeled as Figure 2.74 , to which motion of the cube does the permutation $(12)(34)$ correspond? What about the other permutations of the diagonals?

###### 97.

Find the group of rigid motions of a tetrahedron. Show that this is the same group as $A_4\text{.}$

###### 98.

Prove that $S_n$ is nonabelian for $n \geq 3\text{.}$

Hint

Calculate $(123)(12)$ and $(12)(123)\text{.}$

###### 99.

Show that $A_n$ is nonabelian for $n \geq 4\text{.}$

###### 100.

Prove that $D_n$ is nonabelian for $n \geq 3\text{.}$

###### 101.

Let $\sigma \in S_n$ be a cycle. Prove that $\sigma$ can be written as the product of at most $n-1$ transpositions.

###### 102.

Let $\sigma \in S_n\text{.}$ If $\sigma$ is not a cycle, prove that $\sigma$ can be written as the product of at most $n - 2$ transpositions.

###### 103.

If $\sigma$ can be expressed as an odd number of transpositions, show that any other product of transpositions equaling $\sigma$ must also be odd.

###### 104.

If $\sigma$ is a cycle of odd length, prove that $\sigma^2$ is also a cycle.

###### 105.

Show that a $3$-cycle is an even permutation.

###### 106.

Prove that in $A_n$ with $n \geq 3\text{,}$ any permutation is a product of cycles of length $3\text{.}$

Hint

Consider the cases $(ab)(bc)$ and $(ab)(cd)\text{.}$

###### 107.

Prove that any element in $S_n$ can be written as a finite product of the following permutations.

1. $\displaystyle (1 2), (13), \ldots, (1n)$

2. $\displaystyle (1 2), (23), \ldots, (n- 1,n)$

3. $\displaystyle (12), (1 2 \ldots n )$

###### 108.

Let $G$ be a group and define a map $\lambda_g : G \rightarrow G$ by $\lambda_g(a) = g a\text{.}$ Prove that $\lambda_g$ is a permutation of $G\text{.}$

###### 109.

Prove that there exist $n!$ permutations of a set containing $n$ elements.

###### 110.

Recall that the center of a group $G$ is

\begin{equation*} Z(G) = \{ g \in G : gx = xg \text{ for all } x \in G \}\text{.} \end{equation*}

Find the center of $D_8\text{.}$ What about the center of $D_{10}\text{?}$ What is the center of $D_n\text{?}$

###### 111.

Let $\tau = (a_1, a_2, \ldots, a_k)$ be a cycle of length $k\text{.}$

1. Prove that if $\sigma$ is any permutation, then

\begin{equation*} \sigma \tau \sigma^{-1 } = ( \sigma(a_1), \sigma(a_2), \ldots, \sigma(a_k)) \end{equation*}

is a cycle of length $k\text{.}$

2. Let $\mu$ be a cycle of length $k\text{.}$ Prove that there is a permutation $\sigma$ such that $\sigma \tau \sigma^{-1 } = \mu\text{.}$

Hint

For (a), show that $\sigma \tau \sigma^{-1 }(\sigma(a_i)) = \sigma(a_{i + 1})\text{.}$

###### 112.

For $\alpha$ and $\beta$ in $S_n\text{,}$ define $\alpha \sim \beta$ if there exists an $\sigma \in S_n$ such that $\sigma \alpha \sigma^{-1} = \beta\text{.}$ Show that $\sim$ is an equivalence relation on $S_n\text{.}$

###### 113.

Let $\sigma \in S_X\text{.}$ If $\sigma^n(x) = y$ for some $n \in \mathbb Z\text{,}$ we will say that $x \sim y\text{.}$

1. Show that $\sim$ is an equivalence relation on $X\text{.}$

2. Define the orbit of $x \in X$ under $\sigma \in S_X$ to be the set

\begin{equation*} {\mathcal O}_{x, \sigma} = \{ y : x \sim y \}\text{.} \end{equation*}

Compute the orbits of each element in $\{1, 2, 3, 4, 5\}$ under each of the following elements in $S_5\text{:}$

\begin{align*} \alpha & = (1254)\\ \beta & = (123)(45)\\ \gamma & = (13)(25)\text{.} \end{align*}
3. If ${\mathcal O}_{x, \sigma} \cap {\mathcal O}_{y, \sigma} \neq \emptyset\text{,}$ prove that ${\mathcal O}_{x, \sigma} = {\mathcal O}_{y, \sigma}\text{.}$ The orbits under a permutation $\sigma$ are the equivalence classes corresponding to the equivalence relation $\sim\text{.}$

4. A subgroup $H$ of $S_X$ is transitive if for every $x, y \in X\text{,}$ there exists a $\sigma \in H$ such that $\sigma(x) = y\text{.}$ Prove that $\langle \sigma \rangle$ is transitive if and only if ${\mathcal O}_{x, \sigma} = X$ for some $x \in X\text{.}$

###### 114.

Let $\alpha \in S_n$ for $n \geq 3\text{.}$ If $\alpha \beta = \beta \alpha$ for all $\beta \in S_n\text{,}$ prove that $\alpha$ must be the identity permutation; hence, the center of $S_n$ is the trivial subgroup.

###### 115.

If $\alpha$ is even, prove that $\alpha^{-1}$ is also even. Does a corresponding result hold if $\alpha$ is odd?

###### 116.

If $\sigma \in A_n$ and $\tau \in S_n\text{,}$ show that $\tau^{-1} \sigma \tau \in A_n\text{.}$

###### 117.

Show that $\alpha^{-1} \beta^{-1} \alpha \beta$ is even for $\alpha, \beta \in S_n\text{.}$

###### 118.

Let $r$ and $s$ be the elements in $D_n$ described in Theorem 2.67

1. Show that $srs = r^{-1}\text{.}$

2. Show that $r^k s = s r^{-k}$ in $D_n\text{.}$

3. Prove that the order of $r^k \in D_n$ is $n / \gcd(k,n)\text{.}$