Exercises 4.4 Additional Exercises
1.
Prove that \(\mathbb Z \cong n \mathbb Z\) for \(n \neq 0\text{.}\)
Every infinite cyclic group is isomorphic to \({\mathbb Z}\) by TheoremĀ 4.7.
2.
Prove that \({\mathbb C}^\ast\) is isomorphic to the subgroup of \(GL_2( {\mathbb R} )\) consisting of matrices of the form
Define \(\phi: {\mathbb C}^* \rightarrow GL_2( {\mathbb R})\) by
3.
Prove or disprove: \(U(8) \cong {\mathbb Z}_4\text{.}\)
False.
4.
Prove that \(U(8)\) is isomorphic to the group of matrices
5.
Show that \(U(5)\) is isomorphic to \(U(10)\text{,}\) but \(U(12)\) is not.
6.
Show that the \(n\)th roots of unity are isomorphic to \({\mathbb Z}_n\text{.}\)
Define a map from \({\mathbb Z}_n\) into the \(n\)th roots of unity by \(k \mapsto \cis(2k\pi / n)\text{.}\)
7.
Show that any cyclic group of order \(n\) is isomorphic to \({\mathbb Z}_n\text{.}\)
8.
Prove that \({\mathbb Q}\) is not isomorphic to \({\mathbb Z}\text{.}\)
Assume that \({\mathbb Q}\) is cyclic and try to find a generator.
9.
Let \(G = {\mathbb R} \setminus \{ 1 \}\) and define a binary operation on \(G\) by
Prove that \(G\) is a group under this operation. Show that \((G, *)\) is isomorphic to the multiplicative group of nonzero real numbers.
10.
Show that the matrices
form a group. Find an isomorphism of \(G\) with a more familiar group of order \(6\text{.}\)
11.
Find five nonisomorphic groups of order \(8\text{.}\)
There are two nonabelian and three abelian groups that are not isomorphic.
12.
Prove \(S_4\) is not isomorphic to \(D_{12}\text{.}\)
13.
Let \(\omega = \cis(2 \pi /n)\) be a primitive \(n\)th root of unity. Prove that the matrices
generate a multiplicative group isomorphic to \(D_n\text{.}\)
14.
Show that the set of all matrices of the form
is a group isomorphic to \(D_n\text{,}\) where all entries in the matrix are in \({\mathbb Z}_n\text{.}\)
15.
List all of the elements of \({\mathbb Z}_4 \times {\mathbb Z}_2\text{.}\)
16.
Find the order of each of the following elements.
\((3, 4)\) in \({\mathbb Z}_4 \times {\mathbb Z}_6\)
\((6, 15, 4)\) in \({\mathbb Z}_{30} \times {\mathbb Z}_{45} \times {\mathbb Z}_{24}\)
\((5, 10, 15)\) in \({\mathbb Z}_{25} \times {\mathbb Z}_{25} \times {\mathbb Z}_{25}\)
\((8, 8, 8)\) in \({\mathbb Z}_{10} \times {\mathbb Z}_{24} \times {\mathbb Z}_{80}\)
(a) \(12\text{;}\) (c) \(5\text{.}\)
17.
Prove that \(D_4\) cannot be the internal direct product of two of its proper subgroups.
18.
Prove that the subgroup of \({\mathbb Q}^\ast\) consisting of elements of the form \(2^m 3^n\) for \(m,n \in {\mathbb Z}\) is an internal direct product isomorphic to \({\mathbb Z} \times {\mathbb Z}\text{.}\)
19.
Prove that \(S_3 \times {\mathbb Z}_2\) is isomorphic to \(D_6\text{.}\) Can you make a conjecture about \(D_{2n}\text{?}\) Prove your conjecture.
Draw the picture.
20.
Prove or disprove: Every abelian group of order divisible by \(3\) contains a subgroup of order \(3\text{.}\)
True.
21.
Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order \(6\text{.}\)
22.
Let \(G\) be a group of order \(20\text{.}\) If \(G\) has subgroups \(H\) and \(K\) of orders \(4\) and \(5\) respectively such that \(hk = kh\) for all \(h \in H\) and \(k \in K\text{,}\) prove that \(G\) is the internal direct product of \(H\) and \(K\text{.}\)
23.
Prove or disprove the following assertion. Let \(G\text{,}\) \(H\text{,}\) and \(K\) be groups. If \(G \times K \cong H \times K\text{,}\) then \(G \cong H\text{.}\)
24.
Prove or disprove: There is a noncyclic abelian group of order \(51\text{.}\)
25.
Prove or disprove: There is a noncyclic abelian group of order \(52\text{.}\)
True.
26.
Let \(\phi : G \rightarrow H\) be a group isomorphism. Show that \(\phi( x) = e_H\) if and only if \(x=e_G\text{,}\) where \(e_G\) and \(e_H\) are the identities of \(G\) and \(H\text{,}\) respectively.
27.
Let \(G \cong H\text{.}\) Show that if \(G\) is cyclic, then so is \(H\text{.}\)
Let \(a\) be a generator for \(G\text{.}\) If \(\phi :G \rightarrow H\) is an isomorphism, show that \(\phi(a)\) is a generator for \(H\text{.}\)
28.
Prove that any group \(G\) of order \(p\text{,}\) \(p\) prime, must be isomorphic to \({\mathbb Z}_p\text{.}\)
29.
Show that \(S_n\) is isomorphic to a subgroup of \(A_{n+2}\text{.}\)
30.
Prove that \(D_n\) is isomorphic to a subgroup of \(S_n\text{.}\)
31.
Let \(\phi : G_1 \rightarrow G_2\) and \(\psi : G_2 \rightarrow G_3\) be isomorphisms. Show that \(\phi^{1}\) and \(\psi \circ \phi\) are both isomorphisms. Using these results, show that the isomorphism of groups determines an equivalence relation on the class of all groups.
32.
Prove \(U(5) \cong {\mathbb Z}_4\text{.}\) Can you generalize this result for \(U(p)\text{,}\) where \(p\) is prime?
33.
Write out the permutations associated with each element of \(S_3\) in the proof of Cayley's Theorem.
34.
An automorphism of a group \(G\) is an isomorphism with itself. Prove that complex conjugation is an automorphism of the additive group of complex numbers; that is, show that the map \(\phi( a + bi ) = a  bi\) is an isomorphism from \({\mathbb C}\) to \({\mathbb C}\text{.}\)
35.
Prove that \(a + ib \mapsto a  ib\) is an automorphism of \({\mathbb C}^*\text{.}\)
36.
Prove that \(A \mapsto B^{1}AB\) is an automorphism of \(SL_2({\mathbb R})\) for all \(B\) in \(GL_2({\mathbb R})\text{.}\)
37.
We will denote the set of all automorphisms of \(G\) by \(\aut(G)\text{.}\) Prove that \(\aut(G)\) is a subgroup of \(S_G\text{,}\) the group of permutations of \(G\text{.}\)
38.
Find \(\aut( {\mathbb Z}_6)\text{.}\)
Any automorphism of \({\mathbb Z}_6\) must send 1 to another generator of \({\mathbb Z}_6\text{.}\)
39.
Find \(\aut( {\mathbb Z})\text{.}\)
40.
Find two nonisomorphic groups \(G\) and \(H\) such that \(\aut(G) \cong \aut(H)\text{.}\)
41.
Let \(G\) be a group and \(g \in G\text{.}\) Define a map \(i_g : G \rightarrow G\) by \(i_g(x) = g x g^{1}\text{.}\) Prove that \(i_g\) defines an automorphism of \(G\text{.}\) Such an automorphism is called an inner automorphism. The set of all inner automorphisms is denoted by \(\inn(G)\text{.}\)
42.
Prove that \(\inn(G)\) is a subgroup of \(\aut(G)\text{.}\)
43.
What are the inner automorphisms of the quaternion group \(Q_8\text{?}\) Is \(\inn(G) = \aut(G)\) in this case?
44.
Let \(G\) be a group and \(g \in G\text{.}\) Define maps \(\lambda_g :G \rightarrow G\) and \(\rho_g :G \rightarrow G\) by \(\lambda_g(x) = gx\) and \(\rho_g(x) = xg^{1}\text{.}\) Show that \(i_g = \rho_g \circ \lambda_g\) is an automorphism of \(G\text{.}\) The isomorphism \(g \mapsto \rho_g\) is called the right regular representation of \(G\text{.}\)
45.
Let \(G\) be the internal direct product of subgroups \(H\) and \(K\text{.}\) Show that the map \(\phi : G \rightarrow H \times K\) defined by \(\phi(g) = (h,k)\) for \(g =hk\text{,}\) where \(h \in H\) and \(k \in K\text{,}\) is onetoone and onto.
To show that \(\phi\) is onetoone, let \(g_1 = h_1 k_1\) and \(g_2 = h_2 k_2\) and consider \(\phi(g_1) = \phi(g_2)\text{.}\)
46.
Let \(G\) and \(H\) be isomorphic groups. If \(G\) has a subgroup of order \(n\text{,}\) prove that \(H\) must also have a subgroup of order \(n\text{.}\)
47.
If \(G \cong \overline{G}\) and \(H \cong \overline{H}\text{,}\) show that \(G \times H \cong \overline{G} \times \overline{H}\text{.}\)
48.
Prove that \(G \times H\) is isomorphic to \(H \times G\text{.}\)
49.
Let \(n_1, \ldots, n_k\) be positive integers. Show that
if and only if \(\gcd( n_i, n_j) =1\) for \(i \neq j\text{.}\)
50.
Prove that \(A \times B\) is abelian if and only if \(A\) and \(B\) are abelian.
51.
If \(G\) is the internal direct product of \(H_1, H_2, \ldots, H_n\text{,}\) prove that \(G\) is isomorphic to \(\prod_i H_i\text{.}\)
52.
Let \(H_1\) and \(H_2\) be subgroups of \(G_1\) and \(G_2\text{,}\) respectively. Prove that \(H_1 \times H_2\) is a subgroup of \(G_1 \times G_2\text{.}\)
53.
Let \(m, n \in {\mathbb Z}\text{.}\) Prove that \(\langle m,n \rangle = \langle d \rangle\) if and only if \(d = \gcd(m,n)\text{.}\)
54.
Let \(m, n \in {\mathbb Z}\text{.}\) Prove that \(\langle m \rangle \cap \langle n \rangle = \langle l \rangle\) if and only if \(l = \lcm(m,n)\text{.}\)
55. Groups of order \(2p\).
In this series of exercises we will classify all groups of order \(2p\text{,}\) where \(p\) is an odd prime.
Assume \(G\) is a group of order \(2p\text{,}\) where \(p\) is an odd prime. If \(a \in G\text{,}\) show that \(a\) must have order \(1\text{,}\) \(2\text{,}\) \(p\text{,}\) or \(2p\text{.}\)
Suppose that \(G\) has an element of order \(2p\text{.}\) Prove that \(G\) is isomorphic to \({\mathbb Z}_{2p}\text{.}\) Hence, \(G\) is cyclic.
Suppose that \(G\) does not contain an element of order \(2p\text{.}\) Show that \(G\) must contain an element of order \(p\text{.}\) Hint: Assume that \(G\) does not contain an element of order \(p\text{.}\)
Suppose that \(G\) does not contain an element of order \(2p\text{.}\) Show that \(G\) must contain an element of order \(2\text{.}\)
Let \(P\) be a subgroup of \(G\) with order \(p\) and \(y \in G\) have order \(2\text{.}\) Show that \(yP = Py\text{.}\)
Suppose that \(G\) does not contain an element of order \(2p\) and \(P = \langle z \rangle\) is a subgroup of order \(p\) generated by \(z\text{.}\) If \(y\) is an element of order \(2\text{,}\) then \(yz = z^ky\) for some \(2 \leq k \lt p\text{.}\)
Suppose that \(G\) does not contain an element of order \(2p\text{.}\) Prove that \(G\) is not abelian.
Suppose that \(G\) does not contain an element of order \(2p\) and \(P = \langle z \rangle\) is a subgroup of order \(p\) generated by \(z\) and \(y\) is an element of order \(2\text{.}\) Show that we can list the elements of \(G\) as \(\{z^iy^j\mid 0\leq i \lt p, 0\leq j \lt 2\}\text{.}\)
Suppose that \(G\) does not contain an element of order \(2p\) and \(P = \langle z \rangle\) is a subgroup of order \(p\) generated by \(z\) and \(y\) is an element of order \(2\text{.}\) Prove that the product \((z^iy^j)(z^ry^s)\) can be expressed as a uniquely as \(z^m y^n\) for some non negative integers \(m, n\text{.}\) Thus, conclude that there is only one possibility for a nonabelian group of order \(2p\text{,}\) it must therefore be the one we have seen already, the dihedral group.
56.
Prove that \(\det( AB) = \det(A) \det(B)\) for \(A, B \in GL_2( {\mathbb R} )\text{.}\) This shows that the determinant is a homomorphism from \(GL_2( {\mathbb R} )\) to \({\mathbb R}^*\text{.}\)
57.
Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?

\(\phi : {\mathbb R}^\ast \rightarrow GL_2 ( {\mathbb R})\) defined by
\begin{equation*} \phi( a ) = \begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix} \end{equation*} 
\(\phi : {\mathbb R} \rightarrow GL_2 ( {\mathbb R})\) defined by
\begin{equation*} \phi( a ) = \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \end{equation*} 
\(\phi : GL_2 ({\mathbb R}) \rightarrow {\mathbb R}\) defined by
\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = a + d \end{equation*} 
\(\phi : GL_2 ( {\mathbb R}) \rightarrow {\mathbb R}^\ast\) defined by
\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = ad  bc \end{equation*} 
\(\phi : {\mathbb M}_2( {\mathbb R}) \rightarrow {\mathbb R}\) defined by
\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = b\text{,} \end{equation*}where \({\mathbb M}_2( {\mathbb R})\) is the additive group of \(2 \times 2\) matrices with entries in \({\mathbb R}\text{.}\)
(a) is a homomorphism with kernel \(\{ 1 \}\text{;}\) (c) is not a homomorphism.
58.
Let \(A\) be an \(m \times n\) matrix. Show that matrix multiplication, \(x \mapsto Ax\text{,}\) defines a homomorphism \(\phi : {\mathbb R}^n \rightarrow {\mathbb R}^m\text{.}\)
59.
Let \(\phi : {\mathbb Z} \rightarrow {\mathbb Z}\) be given by \(\phi(n) = 7n\text{.}\) Prove that \(\phi\) is a group homomorphism. Find the kernel and the image of \(\phi\text{.}\)
Since \(\phi(m + n) = 7(m+n) = 7m + 7n = \phi(m) + \phi(n)\text{,}\) \(\phi\) is a homomorphism.
60.
Describe all of the homomorphisms from \({\mathbb Z}_{24}\) to \({\mathbb Z}_{18}\text{.}\)
For any homomorphism \(\phi : {\mathbb Z}_{24} \rightarrow {\mathbb Z}_{18}\text{,}\) the kernel of \(\phi\) must be a subgroup of \({\mathbb Z}_{24}\) and the image of \(\phi\) must be a subgroup of \({\mathbb Z}_{18}\text{.}\) Now use the fact that a generator must map to a generator.
61.
Describe all of the homomorphisms from \({\mathbb Z}\) to \({\mathbb Z}_{12}\text{.}\)
62.
In the group \({\mathbb Z}_{24}\text{,}\) let \(H = \langle 4 \rangle\) and \(N = \langle 6 \rangle\text{.}\)
List the elements in \(HN\) (we usually write \(H + N\) for these additive groups) and \(H \cap N\text{.}\)
List the cosets in \(HN/N\text{,}\) showing the elements in each coset.
List the cosets in \(H/(H \cap N)\text{,}\) showing the elements in each coset.
Give the correspondence between \(HN/N\) and \(H/(H \cap N)\) described in the proof of the Second Isomorphism Theorem.
63.
If \(G\) is an abelian group and \(n \in {\mathbb N}\text{,}\) show that \(\phi : G \rightarrow G\) defined by \(g \mapsto g^n\) is a group homomorphism.
64.
If \(\phi : G \rightarrow H\) is a group homomorphism and \(G\) is abelian, prove that \(\phi(G)\) is also abelian.
Let \(a, b \in G\text{.}\) Then \(\phi(a) \phi(b) = \phi(ab) = \phi(ba) = \phi(b)\phi(a)\text{.}\)
65.
If \(\phi : G \rightarrow H\) is a group homomorphism and \(G\) is cyclic, prove that \(\phi(G)\) is also cyclic.
66.
Show that a homomorphism defined on a cyclic group is completely determined by its action on the generator of the group.
67.
If a group \(G\) has exactly one subgroup \(H\) of order \(k\text{,}\) prove that \(H\) is normal in \(G\text{.}\)
68.
Prove or disprove: \({\mathbb Q} / {\mathbb Z} \cong {\mathbb Q}\text{.}\)
69.
Let \(G\) be a finite group and \(N\) a normal subgroup of \(G\text{.}\) If \(H\) is a subgroup of \(G/N\text{,}\) prove that \(\phi^{1}(H)\) is a subgroup in \(G\) of order \(H \cdot N\text{,}\) where \(\phi : G \rightarrow G/N\) is the canonical homomorphism.
70.
Let \(G_1\) and \(G_2\) be groups, and let \(H_1\) and \(H_2\) be normal subgroups of \(G_1\) and \(G_2\) respectively. Let \(\phi : G_1 \rightarrow G_2\) be a homomorphism. Show that \(\phi\) induces a homomorphism \(\overline{\phi} : (G_1/H_1) \rightarrow (G_2/H_2)\) if \(\phi(H_1) \subset H_2\text{.}\)
71.
If \(H\) and \(K\) are normal subgroups of \(G\) and \(H \cap K = \{ e \}\text{,}\) prove that \(G\) is isomorphic to a subgroup of \(G/H \times G/K\text{.}\)
72.
Let \(\phi : G_1 \rightarrow G_2\) be a surjective group homomorphism. Let \(H_1\) be a normal subgroup of \(G_1\) and suppose that \(\phi(H_1) = H_2\text{.}\) Prove or disprove that \(G_1/H_1 \cong G_2/H_2\text{.}\)
Find a counterexample.
73.
Let \(\phi : G \rightarrow H\) be a group homomorphism. Show that \(\phi\) is onetoone if and only if \(\phi^{1}(e) = \{ e \}\text{.}\)
74.
Given a homomorphism \(\phi :G \rightarrow H\) define a relation \(\sim\) on \(G\) by \(a \sim b\) if \(\phi(a) = \phi(b)\) for \(a, b \in G\text{.}\) Show this relation is an equivalence relation and describe the equivalence classes.