Why care about groups?

This article is a hypothetical conversation between a group theorist (G) and a mathematics student (M) who wants to know about group theory.

Episode 1
M: So what kind of math do you work on?

G: As of now, I'm interested in group theory, particularly finite groups.

M: Group theory? Do people still work in that?

G: Yes, there are a lot of open problems in the subject, particularly the structure theory of finite groups.

M: Oh, I thought all the finite groups had been classified. Wasn't that one of the big achievements around 1980?

G: Actually, no. The classification of finite simple groups, which was completed in 1980, just classifies finite groups with the very special property of being simple, which basically means that the group can't be decomposed in a particular way.

M: Okay, and that doesn't classify finite groups?

G: To take an analogy with natural numbers, simple groups are the analogues of prime numbers. Most numbers aren't prime, so classifying the prime number in no way classifies all numbers.

M: Okay, but you can still decompose any group into simple pieces.

G: True, you can. But it's not that easy. Unlike the case of numbers, where a group is determined by which prime factors it has and to what multiplicity, a finite group isn't determined just by what simple groups it has as factors, and to what multiplicity. Basically, there are many ways of putting the pieces together.

M: Okay, so I guess you people are trying to figure out what are the various ways of putting the pieces together.

G: Not all of us. When the classification was on, that was the big thing in group theory; every group theorist was doing work that could contribute to the classification. But now, group theorists have sort of split off in many different directions, each of which is aimed at collecting more information about groups in a different way.

M: Okay, so group theory doesn't have a Holy Grail now?

G: You could put it that way. The complexity of finite groups is too much for us to expect a classification theorem the way we could expect for in simple groups. But we can keep hoping to understand larger and larger families of groups.

M: I somehow got the impression that group theory was just a whole lot of clever arguments with multiplying stuff and intersecting sets; it doesn't have too many deep ideas the way you have in things like algebraic geometry.

G: Group theory does involve a lot of special cases, but then, so does any other part of mathematics. It's true that in finite group theory we aren't dealing with anything physical or geometric; every finite group does act on geometric things (that falls under the broad header of representation theory) but there's no natural prescription. So group theory really involves a mix of the elementary manipulations and the deeper ideas of representation theory.

M: So you don't think that group theory will ever be tamed completely by the algebraic geometry wave?

G: Group theory is dear to many disciplines of mathematics, including geometric group theory, combinatorial group theory, and algebraic topology. And it has its own distinctive range of ideas and methods. So it's unlikely to get fused into anything else. The interplay with geometric group theory, and group actions in a topological context, is particularly strong.

M: Okay ... I had the impression that group theory's a dead subject, and it's been subsumed into more abstract approaches to math.

G: I guess one can say that group theory is still trying to modernize and functorialize itself. On the other hand, since groups don't have a compelling higher interpretation, that may sometimes make them easier to work with concretely.