Sage

Sage has several convenient functions that will allow us to investigate quickly if a subgroup is normal, and if so, the nature of the resulting quotient group. But for an initial understanding, we can also work with the raw cosets. Let us get our hands dirty first, then learn about the easy way.

The definiton of a factor group requires a normal subgroup, and then we define a way to multiply two cosets of the subgroup to produce another coset. It is important to realize that we can interpret the definition of a normal subgroup to be exactly the condition we need for our new multiplication to be workable. We will do two examples first with a normal subgroup, then with a subgroup that is not normal.

Consider the dihedral group \(D_{8}\) that is the symmetry group of an \(8\)-gon. If we take the element that creates a quarter-turn, we can use it generate a cyclic subgroup of order 4. This will be a normal subgroup (trust us for the moment on this). First, build the (right) cosets (notice there is no output):

So C is a list of lists, with every element of the group G occuring exactly once somewhere. You could ask Sage to print out C for you if you like, but we will try to avoid that here. We want to multiply two cosets (lists) together. How do we do this? Take any element out of the first list, and any element out of the second list and multiply them together (which we know how to do since they are elements of G). Now we have an element of G. What do we do with this element, since we really want a coset as the result of the product of two cosets? Simple we see which coset the product is in. Let us give it a try. We will multiply coset \(1\) with coset \(3\) (there are \(4\) cosets by Lagrange's Theorem). Study the following code carefully to see if you can understand what it is doing, and then read the explanation that follows.

What have we accomplished? In the first line we create p as the product of two group elements, one from coset \(1\) and one from coset \(3\) (C[1], C[3]). Since we can choose any element from each coset, we choose the first element of each (C[ ][0]). Then we count our way through all the cosets, selecting only cosets that contain p. Since p will only be in one coset, we expect a list with just one element. Here, our one-element list contains only 2. So we say the product of coset \(1\) and coset \(3\) is coset \(2\).

Now is a good time to introduce a way to extend Sage and add new functions. We will design a coset-multiplication function. Read the following carefully and then see the subsequent explanation.

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.

You can easily ask Sage if a subgroup is normal or not. This is viewed as a property of the subgroup, but you must tell Sage what the supergroup is, since the answer can change depending on this value. (For example H.is_normal(H) will always be True.) Here are our two examples from above.

The text proves in that \(A_5\) is simple, \(A_5\) has no normal subgroups. We could build every subgroup of \(A_5\) and ask if it is normal in \(A_5\) using the .is_normal() method. But Sage has this covered for us already.

We can also build a quotient group when we have a normal subgroup.

This is useful, but also a bit unsettling. We have the quotient group, but any notion of cosets has been lost, since Q is returned as a new permutation group on a different set of symbols. We cannot presume that the numbers used for the new permutation group Q bear any resemblance to the cosets we get from the .cosets() method. But we can see that the quotient group is described as a group generated by two elements of order two. We could ask for the order of the group, or by Lagrange's Theorem we know the quotient has order \(4\). We can say now that there are only two groups of order four, the cyclic group of order \(4\) and a non-cyclic group of order \(4\), known to us as the Klein \(4\)-group or \({\mathbb Z}_2\times{\mathbb Z}_2\). This quotient group looks like the non-cyclic one since the cyclic group of order 4 has just one element of order 2. Let us see what Sage says.

Yes, that's it.

Finally, Sage can build us a list of all of the normal subgroups of a group. The list of groups themselves, as we have seen before, is sometimes an overwhelming amount of information. We will demonstrate by just listing the orders of the normal subgroups produced.

So, in particular, we see that our quarter-turn subgroup is the only normal subgroup of order \(4\) in this group.