Group Explorer 3.0 Help: Sheets Tutorial

Sheets

Group Explorer since version 2.0 has been able to open multiple group visualizations in one document called a “sheet,” so that they can be compared and homomorphisms between them created, illustrated, and studied.

This page provides a quick introduction to sheets. To go directly to the details of the sheet interface and how to use it, visit this page of the User Manual.

Let’s begin with “Making your own sheets,” but you can jump down to “Getting Group Explorer to make sheets for you,” below, where the fancier stuff shows up.

Making your own sheets

Let’s use a sheet to compare two groups of the same order, $\mathbb{Z}_6$ and $S_3$.

• From the Group Explorer main page, click the Sheet icon on the top right hand side of the page.

  

  This creates a new sheet.

• In the control window on the right hand side of the sheet, select $\mathbb{Z}_6$ from the “Group” drop-down list and click the “Cycle graph” button above it. You should see a small cycle graph for $\mathbb{Z}_6$ appear in the upper left-hand corner of the left side of the sheet. I resized mine slightly and the result was as follows. (To move a visualizer just drag it with the mouse or your finger; to resize it right-click [tap] it and select ‘Resize’ from the resulting context menu. A light blue outline will appear around the visualizer. Then drag starting outside the element in the direction you’d like it to grow. A short click [tap] will dismiss the blue outline. See here for more details.)

• But that’s just one visualizer; we want to have at least two. So let’s repeat the same steps for inserting a cycle graph for $S_3$ as well, but move it a little to the right of the first one, as shown below.

• Now let’s start comparing these groups. In group theory, the means of examining relationships between groups is via homomorphisms. So let’s create one in this sheet, as follows:
  • Right-click [tap] the left cycle graph element to open its context menu
  • Select the “Create Map” option from menu
  • Click [tap] the other cycle graph, the homomorphism target.
  • You should see a new function $f$ appear, connecting the two graphs.

• So far this isn’t very informative, but if we right-click [tap] the morphism label box to open its context menu and select “Edit”, we can do all sorts of interesting things. For instance, you can decide which elements from $\mathbb{Z}_6$ should correspond to which elements from $S_3$. Furthermore, Group Explorer will not let you mess this up (you cannot define a non-homomorphism.) The morphism editing dialog is shown below.

• The homomorphism defaults to the zero map (all elements map to the identity, in this case $e$) but you can change it, of course. I will map $a$ to $r$ and then check the “Draw multiple arrows” box above. The result is the following illustration of one way to map $\mathbb{Z}_6$ to $S_3$.

The arrows require some attention to follow carefully, but you can see how the six-element circle marches around the little three-element circle twice. I’ve taken the liberty of highlighting $\mathbb{Z}_6$ red and its image in $S_3$ red also. (To do so, open the element context menu for the visualizer you’d like to highligh and select “Edit”. Then play with its subsets as documented here.)

This is only the beginning of the potential of sheets. The next section shows much more.

Getting Group Explorer to make sheets for you

The group info pages of Group Explorer are full of links that create sheets. For many common computations, it is very interesting to be able to see the result of the computation visually. I will whet your appetite for such illustrations by giving a few examples here, and providing links for you to browse further yourself.

• To see a short exact sequence exhibiting the normality of a subgroup (and the quotient group it computes):
  • Expand the “Subgroups” section under “Computated Properties” in the group info page.
  • Then find the subgroup in question on the list and follow the link provided.
  • The illustration below shows the normality of $V_4$ in $A_4$.

• To see a lattice of subgroups for a given group:
  • Again, expand the “Subgroups” section of the group info page.
  • Follow the link provided at the top of the resulting page, offering to create a sheet showing the lattice of subgroups.
  • The illustration below shows all subgroups of $S_3$.

• To see the solvable decomposition of any solvable group:
  • Expand the “Solvable” section of the group info page.
  • The decomposition will be reported in text and you can click any of several links on that page to see it illustrated in various ways.
  • The illustration below shows the solvable decomposition for $S_4$.