PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

Exercises3.5Exercises

1.

Find all $$x \in {\mathbb Z}$$ satisfying each of the following equations.
1. $$\displaystyle 3x \equiv 2 \pmod{7}$$
2. $$\displaystyle 5x + 1 \equiv 13 \pmod{23}$$
3. $$\displaystyle 5x + 1 \equiv 13 \pmod{26}$$
4. $$\displaystyle 9x \equiv 3 \pmod{5}$$
5. $$\displaystyle 5x \equiv 1 \pmod{6}$$
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.
1. \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*}
2. \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*}
3. \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*}
4. \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}. \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). \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}. \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}. \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}. \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}. \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 3.2.14: if $$G$$ is a group and $$a, b \in G\text{,}$$ then the equation $$xa = b$$ has a unique solution in $$G\text{.}$$

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 3.3.5 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\}. \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 $$G\text{.}$$
Hint.
Look at $$S_3\text{.}$$

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 \}. \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.
$$1$$

56.

$$C_n\text{,}$$ $$n$$ even.
$$2$$

57.

$$D_n\text{,}$$ $$n$$ odd.
$$n$$

58.

$$D_n\text{,}$$ $$n$$ even.
$$n+1$$

59.

This is an odd-numbered exercise with tasks.

(a)

What is $$1+1\text{?}$$
$$2$$

(b)

(i)
What is $$3+3\text{?}$$
$$6$$
(ii)
What is $$5+5\text{?}$$
$$10$$

60.

This is an even-numbered exercise with tasks.

(a)

What is $$2+2\text{?}$$
$$4$$
What is $$4+4\text{?}$$
$$8$$
What is $$6+6\text{?}$$
$$12$$