Cosets and Lagrange's Theorem

There are several theorems in group theory (and number theory) that are easy to conjecture with enough data.  Group Explorer is an excellent environment in which to gather data that leads students to make conjectures.  We illustrate this with Lagrange's theorem, which states that the order of a subgroup always divides the order of a group.

Here is a small table of contents for this page:

1. The assignment
2. How students can use Group Explorer to gather this data

   	Using Cayley diagrams
   	Using the Multiplication Table

The assignment

Before students come across Lagrange's theorem, the instructor can assign the students experimentation homework in an effort to lead them to conjecture the theorem on their own.  Such an assignment would ask students to load a few groups and find all their subgroups, recording the sizes of each group and its subgroups.  The students should then state what patterns they found.

Such an assignment may be a bit too time-consuming for a classroom or computer lab setting.  Therefore it should be assigned for homework.  Such an assignment requires only beginner-level knowledge of group theory and of Group Explorer, and therefore it may serve well as an introductory assignment.

How students can use Group Explorer to gather this data

Both Cayley diagrams and the Multiplication Table feature are useful for solving this sort of assignment, because both have the facility to highlight subgroups.  We do an example using each method.

Using Cayley diagrams

The group we use for our example is S₃.  Begin by opening the file S_3.gp in Group Explorer.  A rectangular Cayley diagram opens automatically, and you can use that one if you like.  Below, we chose a polar diagram instead.

Right-click on the diagram and choose "Edit Cayley diagram..." from the popup menu.  On the "Define H and a" page, add an element to the subgroup H by choosing an element from the drop-down list and clicking the "Add this element to H" button.  On the "Highlight" page, under the heading "Use node color to highlight," choose the option "the subgroup H."  Click Apply to see the subgroup you've defined highlighted in yellow.

Return to the "Define H and a" page, and experiment with adding and removing various elements, frequently clicking the Apply button to see the subgroups you define highlighted.  A comprehensive list appears below, which a student could make by saving pictures using the File | Save current view... menu item.

Subgroup of S3 generated by < e > This subgroup has 1 element.	Subgroup of S3 generated by < f > This subgroup has 2 elements.
Subgroup of S3 generated by < r > This subgroup has 3 elements.	Subgroup of S3 generated by < fr > This subgroup has 2 elements.
Subgroup of S3 generated by < r, f > This subgroup has 6 elements.	Subgroup of S3 generated by < rf > This subgroup has 2 elements.

Data gathered: The sizes of the subgroups of S₃ are 1, 2, 3, and 6.

Using the Multiplication Table

The group we use for our example is Q₄.  Begin by opening the file Q_4.gp in Group Explorer.  The Multiplication Table opens automatically; bring it to the front so that we can work with it.  The procedure we follow here is the same type of experimentation as in the "Using Cayley diagrams" section above, but the images are different.

Click the "Edit Table" button on the bottom of the Multiplication Table window.  On the "Define H and a" page, add an element to the subgroup H by choosing an element from the drop-down list and clicking the "Add this element to H" button.  On the "Highlight" page, under the heading "Use entire table to highlight," choose the option "the subgroup H."  Click Apply to see the subgroup you've defined highlighted in yellow.

Return to the "Define H and a" page, and experiment with adding and removing various elements, frequently clicking the Apply button to see the subgroups you define highlighted.  A comprehensive list of subgroups appears below, which a student could make by saving pictures using the File | Save current view... menu item.

Subgroup of Q4 generated by < 1 > This subgroup has 1 element.	Subgroup of Q4 generated by < -1 > This subgroup has 2 elements.
Subgroup of Q4 generated by < i > This subgroup has 4 elements.	Subgroup of Q4 generated by < j > This subgroup has 4 elements.
Subgroup of Q4 generated by < k > This subgroup has 4 elements.	Subgroup of Q4 generated by < i, j > This subgroup has 8 elements.

Data gathered: The sizes of the subgroups of Q₄ are 1, 2, 4, and 8.