9.4: Exercises

1

Prove that \(\mathbb Z \cong n \mathbb Z\) for \(n \neq 0\text{.}\)

2

Prove that \({\mathbb C}^\ast\) is isomorphic to the subgroup of \(GL_2( {\mathbb R} )\) consisting of matrices of the form

\[ \begin{pmatrix} a & b \\ -b & a \end{pmatrix}\text{.} \nonumber \]

3

Prove or disprove: \(U(8) \cong {\mathbb Z}_4\text{.}\)

4

Prove that \(U(8)\) is isomorphic to the group of matrices

\[ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\text{.} \nonumber \]

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 \({\mathbb Z}_n\text{.}\)

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

9

Let \(G = {\mathbb R} \setminus \{ -1 \}\) and define a binary operation on \(G\) by

\[ a \ast b = a + b + ab\text{.} \nonumber \]

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

\begin{align*} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \end{align*}

form a group. Find an isomorphism of \(G\) with a more familiar group of order \(6\text{.}\)

11

Find five non-isomorphic groups of order \(8\text{.}\)

12

Prove \(S_4\) is not isomorphic to \(D_{12}\text{.}\)

13

Let \(\omega = \operatorname{cis}(2 \pi /n)\) be a primitive \(n\)th root of unity. Prove that the matrices

\[ A = \begin{pmatrix} \omega & 0 \\ 0 & \omega^{-1} \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \nonumber \]

generate a multiplicative group isomorphic to \(D_n\text{.}\)

14

Show that the set of all matrices of the form

\[ \begin{pmatrix} \pm 1 & k \\ 0 & 1 \end{pmatrix}\text{,} \nonumber \]

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.

1. \((3, 4)\) in \({\mathbb Z}_4 \times {\mathbb Z}_6\)
2. \((6, 15, 4)\) in \({\mathbb Z}_{30} \times {\mathbb Z}_{45} \times {\mathbb Z}_{24}\)
3. \((5, 10, 15)\) in \({\mathbb Z}_{25} \times {\mathbb Z}_{25} \times {\mathbb Z}_{25}\)
4. \((8, 8, 8)\) in \({\mathbb Z}_{10} \times {\mathbb Z}_{24} \times {\mathbb Z}_{80}\)

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.

20

Prove or disprove: Every abelian group of order divisible by \(3\) contains a subgroup of order \(3\text{.}\)

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 \(52\text{.}\)

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 \(H\text{.}\)

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 \(\aut( {\mathbb Z}_6)\text{.}\)

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 one-to-one and onto.

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

\[ \prod_{i=1}^k {\mathbb Z}_{n_i} \cong {\mathbb Z}_{n_1 \cdots n_k} \nonumber \]

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.

1. 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 \(2p\text{.}\)
2. Suppose that \(G\) has an element of order \(2p\text{.}\) Prove that \(G\) is isomorphic to \({\mathbb Z}_{2p}\text{.}\) Hence, \(G\) is cyclic.
3. 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 \(p\text{.}\)
4. Suppose that \(G\) does not contain an element of order \(2p\text{.}\) Show that \(G\) must contain an element of order \(2\text{.}\)
5. Let \(P\) be a subgroup of \(G\) with order \(p\) and \(y \in G\) have order \(2\text{.}\) Show that \(yP = Py\text{.}\)
6. 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 \(2 \leq k \lt p\text{.}\)
7. Suppose that \(G\) does not contain an element of order \(2p\text{.}\) Prove that \(G\) is not abelian.
8. 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 \(\{z^iy^j\mid 0\leq i \lt p, 0\leq j \lt 2\}\text{.}\)
9. 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 non-abelian group of order \(2p\text{,}\) it must therefore be the one we have seen already, the dihedral group.