###### 1

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 3.13 for a short description of the product of groups.)

###### 2

Find all the subgroups of the symmetry group of an equilateral triangle.

###### 3

Compute the subgroups of the symmetry group of a square.

###### 4

Let \(H = \{2^k : k \in {\mathbb Z} \}\text{.}\) Show that \(H\) is a subgroup of \({\mathbb Q}^*\text{.}\)

###### 5

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

###### 6

Let \({\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}\text{.}\) Prove that \({\mathbb T}\) is a subgroup of \({\mathbb C}^*\text{.}\)

###### 7

Let \(G\) consist of the \(2 \times 2\) matrices of the form

\begin{equation*}
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix},
\end{equation*}
where \(\theta \in {\mathbb R}\text{.}\) Prove that \(G\) is a subgroup of \(SL_2({\mathbb R})\text{.}\)

###### 8

Prove that

\begin{equation*}
G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \}
\end{equation*}
is a subgroup of \({\mathbb R}^{\ast}\) under the group operation of multiplication.

###### 9

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

\begin{equation*}
H =
\left\{
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} :
a + d = 0
\right\}.
\end{equation*}
Prove that \(H\) is a subgroup of \(G\text{.}\)

###### 10

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 \(SL_2( {\mathbb R} )\text{.}\)

###### 11

List the subgroups of the quaternion group, \(Q_8\text{.}\)

###### 12

Prove that the intersection of two subgroups of a group \(G\) is also a subgroup of \(G\text{.}\)

###### 13

Prove or disprove: If \(H\) and \(K\) are subgroups of a group \(G\text{,}\) then \(H \cup K\) is a subgroup of \(G\text{.}\)

###### 14

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?

###### 15

Let \(G\) be a group and \(g \in G\text{.}\) Show that

\begin{equation*}
Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \}
\end{equation*}
is a subgroup of \(G\text{.}\) This subgroup is called the **center** of \(G\text{.}\)

###### 16

Let \(a\) and \(b\) be elements of a group \(G\text{.}\) If \(a^4 b = ba\) and \(a^3 = e\text{,}\) prove that \(ab = ba\text{.}\)

###### 17

Give an example of an infinite group in which every nontrivial subgroup is infinite.

Resuming from the Sage exercises in Section 2.4, create the groups `CyclicPermutationGroup(8)` and `DihedralGroup(4)` and name these groups `C` and `D`, respectively.

######
18Sage Exercise 1

For `C` locate the one subgroup of order \(4\text{.}\) The group `D` has three subgroups of order \(4\text{.}\) Select one of the three subgroups of `D` that has a different structure than the subgroup you obtained from `C`.

The `.subgroups()` method will give you a list of all of the subgroups to help you get started. A Cayley table will help you tell the difference between the two subgroups. What properties of these tables did you use to determine the difference in the structure of the subgroups?

######
19Sage Exercise 2

The `.subgroup(elt_list)` method of a group will create the smallest subgroup containing the specified elements of the group, when given the elements as a list `elt_list`. Use this command to discover the shortest list of elements necessary to recreate the subgroups you found in the previous exercise. The equality comparison, `==`, can be used to test if two subgroups are equal.