###### 1.

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

Hint

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

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

Define $\phi: {\mathbb C}^* \rightarrow GL_2( {\mathbb R})$ by

\begin{equation*} \phi(a + bi) = \begin{pmatrix} a & b \\ -b & a \end{pmatrix}\text{.} \end{equation*}
###### 3.

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

Hint

False.

###### 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 ${\mathbb Z}_n\text{.}$

Hint

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 ${\mathbb Z}\text{.}$

Hint

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

\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 $8\text{.}$

Hint

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

\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 ${\mathbb Z}_{10} \times {\mathbb Z}_{24} \times {\mathbb Z}_{80}$

Hint

(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.

Hint

Draw the picture.

###### 20.

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

Hint

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 $52\text{.}$

Hint

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 $H\text{.}$

Hint

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

Hint

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 one-to-one and onto.

Hint

To show that $\phi$ is one-to-one, 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

\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.

###### 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?

1. $\phi : {\mathbb R}^\ast \rightarrow GL_2 ( {\mathbb R})$ defined by

\begin{equation*} \phi( a ) = \begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix} \end{equation*}
2. $\phi : {\mathbb R} \rightarrow GL_2 ( {\mathbb R})$ defined by

\begin{equation*} \phi( a ) = \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \end{equation*}
3. $\phi : GL_2 ({\mathbb R}) \rightarrow {\mathbb R}$ defined by

\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = a + d \end{equation*}
4. $\phi : GL_2 ( {\mathbb R}) \rightarrow {\mathbb R}^\ast$ defined by

\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = ad - bc \end{equation*}
5. $\phi : {\mathbb M}_2( {\mathbb R}) \rightarrow {\mathbb R}$ 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 ${\mathbb R}\text{.}$

Hint

(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 $\phi\text{.}$

Hint

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 ${\mathbb Z}_{18}\text{.}$

Hint

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 $N = \langle 6 \rangle\text{.}$

1. List the elements in $HN$ (we usually write $H + N$ for these additive groups) and $H \cap N\text{.}$

2. List the cosets in $HN/N\text{,}$ showing the elements in each coset.

3. List the cosets in $H/(H \cap N)\text{,}$ showing the elements in each coset.

4. 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.

Hint

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 $G_1/H_1 \cong G_2/H_2\text{.}$

Hint

Find a counterexample.

###### 73.

Let $\phi : G \rightarrow H$ be a group homomorphism. Show that $\phi$ is one-to-one 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.