##
Exercises 1.5 Exercises

### 1.

Find all \(x \in {\mathbb Z}\) satisfying each of the following equations.

\(\displaystyle 3x \equiv 2 \pmod{7}\)

\(\displaystyle 5x + 1 \equiv 13 \pmod{23}\)

\(\displaystyle 5x + 1 \equiv 13 \pmod{26}\)

\(\displaystyle 9x \equiv 3 \pmod{5}\)

\(\displaystyle 5x \equiv 1 \pmod{6}\)

\(\displaystyle 3x \equiv 1 \pmod{6}\)

## Hint.

(a) \(3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\text{;}\) (c) \(18 + 26 \mathbb Z\text{;}\) (e) \(5 + 6 \mathbb Z\text{.}\)

### 2.

Which of the following multiplication tables defined on the set \(G = \{ a, b, c, d \}\) form a group? Support your answer in each case.

\begin{equation*}
\begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & c & d & a \\ b & b & b & c & d \\ c & c & d & a & b \\ d & d & a & b & c \end{array}
\end{equation*}

\begin{equation*}
\begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & d & c \\ c & c & d & a & b \\ d & d & c & b & a \end{array}
\end{equation*}

\begin{equation*}
\begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & c & d & a \\ c & c & d & a & b \\ d & d & a & b & c \end{array}
\end{equation*}

\begin{equation*}
\begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & c & d \\ c & c & b & a & d \\ d & d & d & b & c \end{array}
\end{equation*}

## Hint.

(a) Not a group; (c) a group.

### 3.

Write out Cayley tables for groups formed by the symmetries of a rectangle and for \(({\mathbb Z}_4, +)\text{.}\) How many elements are in each group? Are the groups the same? Why or why not?

### 4.

Describe the symmetries of a rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a rectangle and the symmetries of a rhombus. Are the symmetries of a rectangle and those of a rhombus the same?

### 5.

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 \(D_4\text{.}\)

### 6.

Give a multiplication table for the group \(U(12)\text{.}\)

## Hint.

\begin{equation*}
\begin{array}{c|cccc} \cdot & 1 & 5 & 7 & 11 \\ \hline 1 & 1 & 5 & 7 & 11 \\ 5 & 5 & 1 & 11 & 7 \\ 7 & 7 & 11 & 1 & 5 \\ 11 & 11 & 7 & 5 & 1 \end{array}
\end{equation*}

### 7.

Let \(S = {\mathbb R} \setminus \{ -1 \}\) and define a binary operation on \(S\) by \(a \ast b = a + b + ab\text{.}\) Prove that \((S, \ast)\) is an abelian group.

### 8.

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.

### 9.

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

### 10.

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*}

### 11.

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 \(AB \in GL_2({\mathbb R})\text{.}\)

### 12.

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.

### 13.

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

### 14.

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.

### 15.

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

## Hint.

There is a nonabelian group containing six elements.

### 16.

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.

### 17.

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

## Hint.

The are five different groups of order 8.

### 18.

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 \(a_2\text{,}\) \(\ldots\text{,}\) 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.

### 19.

Show that

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

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

### 20.

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*}

### 21.

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*}

### 22.

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 \(n\text{.}\)

### 23.

Show that addition and multiplication mod \(n\) are associative operations.

### 24.

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

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

### 25.

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*}

### 26.

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 \(k \neq 1\text{.}\)

### 27.

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

### 28.

Prove the remainder of

Proposition 1.2.14: if

\(G\) is a group and

\(a, b \in G\text{,}\) then the equation

\(xa = b\) has a unique solution in

\(G\text{.}\)
### 29.

### 30.

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 \(a,
b, c \in G\text{.}\)

### 31.

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 \(ba = ab\text{.}\)

### 32.

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 \(a^2 = e\text{.}\)

### 33.

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.

### 34.

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 1.3.5 for a short description of the product of groups.)

### 35.

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

## Hint.

\(H_1 = \{ id \}\text{,}\) \(H_2 = \{ id, \rho_1, \rho_2 \}\text{,}\) \(H_3 = \{ id, \mu_1 \}\text{,}\) \(H_4 = \{ id, \mu_2 \}\text{,}\) \(H_5 = \{ id, \mu_3 \}\text{,}\) \(S_3\text{.}\)

### 36.

Compute the subgroups of the symmetry group of a square.

### 37.

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

### 38.

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 \(\mathbb{Z}\text{.}\)

### 39.

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

### 40.

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

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

### 41.

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, \((a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}\)

### 42.

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 \(G\text{.}\)

### 43.

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 \(SL_2( {\mathbb R} )\text{.}\)

### 44.

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

### 45.

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

### 46.

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

### 47.

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?

### 48.

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 \(G\text{.}\)

### 49.

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

## Hint.

Since \(a^4b = ba\text{,}\) it must be the case that \(b = a^6 b = a^2 b a\text{,}\) and we can conclude that \(ab = a^3 b a = ba\text{.}\)

### 50.

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

### 51.

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

### 52.

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

### 53.

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 \(G\text{.}\)

### 54.

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

### Exercise Group.

In each group, how many solutions are there to \(x^2=e\text{?}\)

#### 55.

\(C_n\text{,}\) \(n\) odd.

#### 56.

\(C_n\text{,}\) \(n\) even.

#### 57.

\(D_n\text{,}\) \(n\) odd.

#### 58.

\(D_n\text{,}\) \(n\) even.

### 59.

This is an odd-numbered exercise with tasks.

#### (a)

What is \(1+1\text{?}\)

#### (b)

##### (i)

What is \(3+3\text{?}\)

##### (ii)

What is \(5+5\text{?}\)

### 60.

This is an even-numbered exercise with tasks.

#### (a)

What is \(2+2\text{?}\)

#### (b)

##### (i)

What is \(4+4\text{?}\)

##### (ii)

What is \(6+6\text{?}\)