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Exercises 8.4 Exercises

1.

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*}

2.

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?

3.

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

4.

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 \(D_4\text{.}\)

5.

Give a multiplication table for the group \(\mathbb Z_{12}^\times\text{.}\)

6.

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.

7.

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

8.

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

9.

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 is known as the Heisenberg group and 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*}

10.

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

11.

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.

12.

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

13.

Given the groups \({\mathbb R}^{\times}\) and \({\mathbb Z}^+\text{,}\) let \(G = {\mathbb R}^{\times} \times {\mathbb Z}^+\text{.}\) 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.

14.

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

15.

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

16.

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

17.

Show that there are \(n!\) permutations of a set containing \(n\) items.

18.

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

19.

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

20.

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

21.

Prove the remainder of Proposition 8.18: if \(G\) is a group and \(a, b \in G\text{,}\) then the equation \(xa = b\) has a unique solution in \(G\text{.}\)

22.

Prove Theorem 8.20.

23.

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

24.

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

25.

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

26.

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.

27.

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

28.

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

29.

Compute the subgroups of the symmetry group of a square.

30.

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

31.

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

32.

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

33.

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}, \end{equation*}

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

34.

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 \({\mathbb R}^{\times}\) under the group operation of multiplication.

35.

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 \(G\text{.}\)

36.

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

37.

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

38.

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

39.

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

40.

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?

41.

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 \(G\text{.}\)

42.

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

43.

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

44.

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

45.

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

46.

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 \(G\text{.}\)

47.

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 \(G\text{.}\)