$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$

## Section3.8Sage 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 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.