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Exercises 1.5 Exercises

1.

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

(a)

\(3x \equiv 2 \pmod{7}\)
Hint.
\(3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\)

(b)

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

(c)

\(5x + 1 \equiv 13 \pmod{26}\)
Hint.
\(18 + 26 \mathbb Z\)

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.

(a)

\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*}
Hint.
Not a group.

(b)

\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*}

(c)

\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*}
Hint.
A group.

(d)

\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*}

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 1.2.14: if \(G\) is a group and \(a, b \in G\text{,}\) then the equation \(xa = b\) has a unique solution in \(G\text{.}\)

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\}. \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 \(x^2=e\text{?}\)
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