In his Contemporary Abstract Algebra text, Gallian asserts that Sylow's Theorem(s) and Lagrange's Theorem are the two most important results in finite group theory. He also provides this quote by G.A. Miller:
Generally these three results are implied by the expression "Sylow's Theorem." All of them are of fundamental importance. In fact, if the theorems of group theory were arranged in order of their importance Sylow's Theorem might reasonably occupy the second place – coming next to Lagrange's Theorem in such an arrangement."
The notion of 'most important theorem' in any area of mathematics is of course highly subjective, but if we depart from this phrasing and think of theorems that are very far-reaching, widely applicable, or theorems that impacted group theory the most (surely there is a good deal of overlap), do Lagrange's and Sylow's theorems still top this list when we consider the whole of group theory? If not, what theorem or theorems would?
What theorem(s) within strictly infinite group theory would top such a list?
Best Answer
What is the most important theorem in infinite group theory? I would say that it is the rather annoying theorem which says that "groups are (very!) complicated". There are three fundamental theorems on "complicated-ness", and I will mention each of these below. However, the proofs of the other two are based on the first, which is why I am proposing it for the "most important" theorem. The three problems I will talk about are called Dehn's problems, and were posed by Dehn in 1914. He proved that they are soluble for the fundamental groups of closed, orientable surfaces of dimension two. The key point is that they are not soluble in general. (Note that there are plenty of similar results, but these three are the classical, fundamental ones.)
A general reference for most of the stuff below is the paper Decision problems for groups: surveys and reflections by Chuck Miller III. You can download this from his webpage. Another reference is the book of Lyndon-Schupp entitled Combinatorial group theory (this also provides a reference for my mentioning of Dehn's proof that the problems are soluble for the fundamental groups of closed, orientable surfaces of dimension two). [Note. Since writing this post I came across an answer of Henry Wilton on MathOverflow. See here. It is very relevant to this post, and I believe explains extremely well why we as geometric group theorists care about the existence an non-existence of algorithms.]
Presentations. Before we can state the theorems, we need to briefly introduce presentations. Let $G$ be a group. Then every group can be given by a presentation, so $G=\langle X; R\rangle$. Here, $X$ is a set of generators for $G$ and $R$ is a set of relators which tell you how the generators multiply together. If $G$ can be given by a presentation such that $X$ and $R$ are both finite then $G$ is said to be finitely presentable. For example, the Klein-four group $V$ is given by the following presentation. $$\langle a, b; a^2=1, b^2=1, ab=ba\rangle$$ Therefore, $V$ is finitely presentable (if I recall correctly, this is a worked example at the start of the book Combinatorial group theory by Magnus-Karrass-Solitar (note: not Lyndon-Schupp - same name, different books)). Indeed, every finite group is finitely presentable as one can take $X$ to be all the (non-trivial) elements of $G$ and then take the elements of $R$ to be simply the way that two elements multiply together, so if $g, h, k\in G$ such that $gh=k$ then $ghk^{-1}\in R$. For example, the Klein-four group can be given by the following presentation. $$V\cong\langle a, b, c; ab=c, ab=b, a^2=1, ba=c, b^2=1, bc=1, ca=b, cb=a, c^2=1\rangle$$ Note that there are infinite groups with finite presentations, for example $\langle a;-\rangle$ and $\langle a, b; b=1\rangle$ are two different presentations of the infinite cyclic group.
Algorithms. It is not necessary, but you might find this post of Arturo Magidin helpful before continuing. Or perhaps you might want to read it afterwards. It discusses what it means for things to be "decidable", and so on.
The word problem for groups. One of the most natural questions one can ask when given a group via a (finite or otherwise) presentation is the following.
That is, if $G=\langle X; R\rangle$ and $U(X)$, $V(X)$ are words over $X$, is it true that $U(X)=_GV(X)$? Equivalently, "Is it true that $U(X)V^{-1}(X)=_G1$?", and so the word problem for groups asks this.
It turns out that this is an intrinsic property of the group, linked to Turing machines. But that is not the theorem I want to tell you about (it just lets me talk about groups as opposed to presentations). The theorem I want to tell you about is as follows.
Or more informally, there exists a group $G$ such that it is impossible to tell if an element is trivial or not. This was proven in the 1950s. And should blow your mind!
Note that the word problem is soluble for finite groups.
The conjugacy problem for groups. Let $G$ be given by a presentation $\langle X;R\rangle$. Then the conjugacy problem for groups asks if when given two elements $U(X)$ and $V(X)$ it is decidable whether these elements are conjugate, that is, whether there exists a third element $W(X)$ such that $W^{-1}UW=V$. Note that insoluble word problem implies insoluble conjugacy problem. However, there is a middle way.
The isomorphism problem for groups. Let $\langle X; R\rangle$ and $\langle Y; S\rangle$ be two presentations. Then can we decide if they define isomorphic groups? Erm, once again, no...
Actually, Adian-Rabin's result is deeper. They prove that "Markov properties of finitely presented groups are not recursively recognisable". This is in Miller's paper.
Why Novikov-Boone? Now, all three results are impressive. However, I am proposing the Novikov-Boone result as the "most important" because one can prove the other results using it. It is therefore the most fundamental of the three theorems. A rather lovely example of this is the paper Isomorphism versus commensurability for a class of finitely presented groups by Arzhantseva-Lafont-Minasyan, which can be found here. The use Novikov-Boone to give groups with insoluble isomorphism problem in an exceptionally elegant way. The insolubility of lots of other decision problems for groups follow from Novikov-Boone. Arzhantseva-Lafont-Minasyan's paper includes other examples, but one can derive using rather elementary means that it is insoluble in general whether two group homomorphisms are the same.
What next? Another reason for proposing this theorem is that finding classes of groups which have soluble word problem is more than just a fun game: it is a major driving force in geometric group theory. For example, Dehn started with closed, orientable surfaces of dimension two. This was generalised to "small cancellation theory" (Greendlinger, Lyndon, etc.) which led to "hyperbolic groups" (Gromov), and this was all motivated by soluble word and conjugacy problems. In time, this theory led to Wise's work on groups with a "quasi-convex hierarchy", which has led to the resolution of the virtually Haken conjecture by Agol. (The virtually Haken conjecture is an important now-theorem in 3-manifold theory.)