Abstract Algebra: Theory and Applications, Remixed!

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 \(AB \neq BA\text{.}\)

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

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

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

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.

10.

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

11.

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

12.

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

13.

Show that

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

14.

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

15.

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

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 \(n\text{:}\)

19.

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

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 \(G\text{.}\)

23.

Prove Theorem 2.23 .

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.

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.

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

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

35.

Prove that

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

36.

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

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 \(G\text{.}\)

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

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

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

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.

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. \(\displaystyle 312 \in {\mathbb Z}_{471}\)

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 \((1 + \sqrt{3}\, i) / 2\)

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. \(\displaystyle \displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}\)

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. \(\displaystyle {\mathbb R}^\ast\)

59.

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

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

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. \(\displaystyle 971^{ 321} \pmod{ 765}\)

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 \(\langle g \rangle \cap \langle h \rangle \text{?}\)

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 \(G\text{.}\)

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.

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

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

92.

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

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 \(n \geq 3\text{.}\)

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 \(3\text{.}\)

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

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 \(\sigma \tau \sigma^{-1 } = \mu\text{.}\)

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 \(n / \gcd(k,n)\text{.}\)