## Exercises 1.7 Sage Exercises

These exercises are about becoming comfortable working with groups in Sage.

### 1.

Create the groups `CyclicPermutationGroup(8)`

and `DihedralGroup(4)`

and name these groups `C`

and `D`

, respectively. We will understand these constructions better shortly, but for now just understand that both objects you create are actually groups.

### 2.

Check that `C`

and `D`

have the same size by using the `.order()`

method. Determine which group is abelian, and which is not, by using the `.is_abelian()`

method.

### 3.

Use the `.cayley_table()`

method to create the Cayley table for each group.

### 4.

Write a nicely formatted discussion identifying differences between the two groups that are discernible in properties of their Cayley tables. In other words, what is {\em different} about these two groups that you can “see” in the Cayley tables? (In the Sage notebook, a Shift-click on a blue bar will bring up a mini-word-processor, and you can use use dollar signs to embed mathematics formatted using TeX syntax.)

### 5.

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?

### 6.

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.