Recap - Subgroups

Subgroups are an important part of group theory. I want to review a little bit of the last few posts and try to tie things together a little bit.

A subset is simply a group that's inside of another group. If G is a group and H is a subset of G, then H is a subgroup if H is a group under the operation of G. If G is a group and we're given a subset, H, of G, we learned that there are three test to determine if H is a subgroup of G - the one-step subgroup test, the two-step subgroup test, and the finite subgroup test. One thing that these subgroup tests don't give us is a way to find subgroups inside of a group if we're not given a subset to start with. We did see a couple of ways, though, to find subgroups within any arbitrary group, including the following:

• If G is a group, then Z(G) is a subgroup of G.
• If G is a group, then C(a) is a subgroup of G ∀a∈G.
• If G is a group, then <a> is a subgroup of G ∀a∈G.

We also looked at some particular examples of subgroups. We saw that 2Z is a subgroup of Z. In fact, it is the case that if nZ = {nk : k∈Z} then nZ is a subgroup of Z. It also happens to be the case that nZ = <n>. We also looked at some examples of cyclic subgroups, namely <2> and <3> in Z(6).

As I alluded to before, subgroups are a very important part of group theory - especially the theory of finite groups. In a perfect world, the ultimate goal of group theory would be to be able to describe every single group that there could possibly be. (Note that I said "could possibly be," and not "is." I'm not going to explain it now, but this is a very interesting distinction.) From what I hear, this is an extremely unrealistic goal and will not be completed in my grandchildren's lifetime. However, we do have a small glimpse into this endeavor. There is something called a normal subgroup (which we will learn about eventually - we've got a bit of ground to cover first) and a group that has no normal subgroups is called simple. In arguably the greatest mathematical discovery of the last 25 years, we do know every single imaginable finite simple group. The proof is a series of papers that spans literally 10,000 pages and maybe 10 people in the whole world have read and understood the entire thing, but still, its been done. And if an arbitrary group is a molecule, then simple groups are its atoms - that is to say that every group can be made (in math language we say "is isomorphic to") by combining some of its simple subgroups. So, every imaginable group can be constructed from these finite simple subgroups that we already understand, which is very, very cool.

Before I go, there's probably one last thing I should explain. You might be wondering, if we know all the finite simple groups, and every group can be constructed from finite simple groups, then why is it that we don't know all of the groups? Well, the answer is that even though we know all the atoms (you know, to a reasonable degree), we can't know all the molecules because we don't know all the imaginable ways to construct them. Similarly, in group theory, although we know all the building blocks, we don't know all possible ways of combining them.

References

Previous Related Posts: Center and Centralizers