*Please read all of the instructions before getting started.*

## instructions

This lab should be completed in groups of 1—4. You do not need to be in the same group as your presentation partner. I recommend keeping this window open as a tab in your browser.

First, one member of your group will need to log into the Sage Notebook server. Go to http://sagenb.org (Sage recommends that you use Firefox). Once you are logged into the Sage Notebook server, you will be in your "`Home`" directory. Click on "`New Worksheet`" towards the top left. A small window will pop up asking you what you want to call the worksheet. Name your worksheet

`MA4140 Lab 2 (name-1, name-2, etc.)`

where you should replace

`name-1`,

`name-2`, etc. with the appropriate first names of your group members separated by commas. You will be presented with a blank Sage worksheet.

Before you actually do anything with this worksheet, I want you to share it with me and the other members of your group. Click "`Save & Quit`" in the upper right hand corner. This will return you to your "`Home`" directory. Now, click on "`Share now`" in the middle column for your newly created worksheet. As you did on Sage Lab 1, type in the Sage usernames of the members of your group (separated by commas). If you do not know the usernames of your group members, you can get them here. You should also share the worksheet with me (my Sage username is `dcernst`). Once you have typed in the appropriate usernames, click "`Invite Collaborators`".

Now, open the worksheet back up. The goal of this lab is to explore two different groups of order 6.

In a Sage cell, type

G=CyclicPermutationGroup(6)

and evaluate the cell. It may look like nothing happened, but you just told Sage that you want to call the `CyclicPermutationGroup` with 6 elements $G$. It turns out that this group is really just $\mathbb{Z}_6$, but don't worry about the details of this right now. All you need to know is that we have a group called $G$.

Now, in the next Sage cell, type

H=SymmetricGroup(3)

and click "`evaluate`". Again, you have just named a group $H$, which happens to be the group $D_3$ (symmetry group for an equilateral triangle) that we've discussed.

I want you to use Sage to verify the order of $G$ and $H$ using the `.order()` command. In an empty Sage cell, type

G.order()

and evaluate. Next, do the same thing for $H$. Below the output you just got, `Shift+click` on a purple line to open the mini text editor and answer the following question:

(1) What do you observe about the orders of $G$ and $H$?

Next, I want you to use Sage to check whether each group is abelian or not using the `.is_abelian()` command. In an empty Sage cell, type

G.is_abelian()

and evaluate. Do the same for $H$. `Shift+click` on a purple line to open a text editor and answer the following question:

(2) Which of the two groups are abelian?

Now, I want you to use Sage to create the Cayley tables for each group using the `.cayley_table()` command. In an empty Sage cell, type

G.cayley_table()

and evaluate. Do the same thing for $H$. Now, using the tables for each group, I want you to answer the following questions (for each group):

(3) What is the identity?

(4) Are there any elements that are their own inverse? If so, what are they?

(5) Which elements commute with every other element?

(6) For each group, list at least one non-trivial subgroup.

Lastly, answer the following question.

(7) Can these groups be the same group? Explain your answer and give **three** distinct reasons.

## getting help

If you need help, you can always contact me, or even better, post a question in the forum. In addition, there are some useful links on the Sage help page.

## due date

This lab is due by **5PM on Friday, March 5**.