## Exercises9.4Exercises

###### 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{equation*} \begin{pmatrix} a & b \\ -b & a \end{pmatrix}\text{.} \end{equation*}
###### 3.

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

###### 4.

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

\begin{equation*} \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{.} \end{equation*}
###### 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

\begin{equation*} a \ast b = a + b + ab\text{.} \end{equation*}

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

\begin{equation*} A = \begin{pmatrix} \omega & 0 \\ 0 & \omega^{-1} \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{equation*}

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

###### 14.

Show that the set of all matrices of the form

\begin{equation*} \begin{pmatrix} \pm 1 & k \\ 0 & 1 \end{pmatrix}\text{,} \end{equation*}

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

\begin{equation*} \prod_{i=1}^k {\mathbb Z}_{n_i} \cong {\mathbb Z}_{n_1 \cdots n_k} \end{equation*}

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.