Theorems with the greatest impact on group theory as a whole

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?

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.

"Can I decide if two words in the generators are equal?"

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.

Let $G$ be a group given by a presentation $\langle X; R\rangle$. Then, given a word $W(X)$, is it decidable if $W(X)=_G1$?

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.

Possibly the most important theorem in (infinite) group theory.

Theorem (Novikov-Boone) There is a finitely presented group $G$ such that the word problem in $G$ is undecidable.

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.

Theorem (Collins) There exists a finitely presented group $G$ which has soluble word problem but insoluble conjugacy problem.

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...

Theorem (Adian-Rabin) It is undecidable whether a finite presentation defines the trivial group.

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.)

A natural answer to the question in your title is the Classification. This does not quite address what you ask in the body of the question, as it is a result strictly within finite group theory, but I believe it is important enough, and of a different nature than such basic pillars as the Sylow theorems or Lagrange, that it deserves some remarks. I address extensions to infinite group theory at the end.

Let me quote from a nice paper by Michael Aschbacher, one of the key figures in the classification effort. Particularly, note the second point. (Emphasis is mine.)

Conventional wisdom says the ideal mathematical proof should be short, simple and elegant. However, there are now examples of very long, complicated proofs, and as mathematics continues to mature, more examples are likely to appear.

I have some experience with one such effort: the Classification of the finite simple groups. [...] I'll begin by listing some features of the theorem and its proof. [...]

First, the proof of the Classification is very long and complicated. As a guess, the proof involves perhaps 10 000 pages in hundreds of papers, written by hundreds of mathematicians. [...]

Second, the theorem is very useful. One cannot do serious finite group theory without the Classification, and it has made possible numerous applications of finite group theory in other branches of mathematics. One can speculate that a proof of the complexity of the Classification would be unlikely to evolve in the absence of such strong incentives. One can also speculate that such theorems can only be proved via some kind of evolutionary process: the extent of the problem and possible paths to a solution only become visible after a large amount of preliminary investigation and experimentation.

Third, at first glance the Classification is a prototypical classification theorem. [...]

But also fourth, the collection $L$ of examples is large, varied, and of great interest, and each member has a rich structure. [...] [The] proof supplies a wealth of detailed information about the structure of members of $L$. Such information is a prerequisite for applying the Classification. Thus after a bit more thought, the Classification is more than just a 'classification theorem'.

Fifth, the proof is inductive and depends upon a good knowledge of the structure of members of $L$. [...]

Now I'd like to draw some implications from the example. I began with the observation that the ideal proof is short, simple, and elegant. The proof in our example has none of these desirable qualities. That hasn't stopped mathematicians from appealing to the theorem, but it does raise various questions.

First, because of the complexity of the proof and the absence of a definitive treatment in the literature, one can ask if the theorem has really been proved. After all, the probability of an error in the proof is one. Indeed, presumably any attempt to write down a proof of such a theorem must contain many mistakes. Human beings are not capable of writing up a 10 000 page argument which is entirely free of errors. Thus if we demand that our proofs be error free, then the Classification can't be proved via techniques currently available. [...]

By concensus [sic] of the community of group theorists, the Classification has been accepted as a theorem for roughly 25 years, despite the fact that, for at least part of that period, gaps in the proof were known to exist. At this point in time, all known gaps have been filled. [...]

I believe the Classification is an example of mathematics coming to grips with a complex information rich problem. [...]

[T]he utility of the theorem stems from two facts: First, it seems to be possible to reduce most questions about finite groups to questions about simple groups. Second, the explicit description of the groups on the list $L$ supplied by very effective representations of most of the groups, make it possible to obtain a vast amount of detailed information about the groups. [...]

My guess is that we will begin to encounter many more such problems, theorems, and proofs in the near future.

The paper is Highly complex proofs and implications of such proofs, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 363 (1835), (2005), 2401–2406. MR2197656 (2006h:00004). It appeared on a volume on The nature of mathematical proof.

Finally, let me mention that there is ongoing work by model theorists to extend the results and techniques of (parts of) the classification to appropriate infinite settings, hoping that this will eventually prove similarly useful in certain contexts. Rather than studying arbitrary simple groups, the effort is currently centered on those simple groups that have finite Morley rank, a certain generalization of the idea of dimension. This finite dimensionality assumption has proved very fruitful in terms of lifting and generalizing techniques from the strictly finite setting.

Though many researchers are involved in this program, let me highlight Tuna Altınel, Alexandre V. Borovik, and Gregory Cherlin, who have co-authored a monograph on this topic, Simple groups of finite Morley rank, Mathematical Surveys and Monographs, 145. American Mathematical Society, Providence, RI, 2008. MR2400564 (2009a:20046).

This page has additional details on the goals of the program, and further references.

I quite like this very old question, and I think there is room for another answer.

I'll propose the theorem underlying Dehn's Algorithm as having one of the greatest impacts on infinite group theory:

Theorem: For $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface, with generators $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ and with defining relation $[a_1,b_1]\cdots[a_g,b_g]=1$. Let $w$ be a nontrivial reduced word in the generators $S_g$ such that $w$ represents the identity in $\Gamma_g$. Then $w$ contains a subword $r_1$ of length $k \geq 2g+1$ such that there exists a word $r_2$ of length $4g-k$ with $r_1 r_2$ a cyclic permutation of $[a_1,b_1]\cdots[a_g,b_g]$.

Now that may sound technical, but here's the immediate point. Regarding Dehn's "Word Problem" which was addressed in the answer of @user1729, the theorem above represents Dehn's example of a positive solution to the Word Problem (just as the Novikov-Boone Theorem proposed by @user1729 represents a negative solution): applying that theorem immediately leads to an algorithmic solution of the word problem in $\Gamma_g$.

But beyond that, the method of proof used by Dehn, and in particular his use of the geometry of the hyperbolic plane, was the motivation behind a couple of tremendous explosions in the progress of infinite group theory.

The first, long wave of that explosion is represented by "combinatorial group theory" and associated work in "small cancellation theory". For example, Dehn's Algorithm has the simple corollary that the isoperimetric function or Dehn function of $\Gamma_g$ has a linear upper bound: the number of relators needed to prove that a word does or does not represent the identity is bounded above by a linear function of the word length. One way to think of small cancellation theory is that it was devoted to discovering broadly formulated, new conditions on groups under which nice upper bounds of the Dehn function could be derived.

A second explosion grew out of several works which would not have been possible without the groundwork laid by Dehn's algorithm and combinatorial group theory, those works being: Milnor's theorems on growth functions of groups, and Gromov's theorem on groups of polynomial growth which answered one of Milnor's questions; Stallings' ends theorem; and the Mostow rigidity theorems regarding lattices in semisimple Lie groups. What these things triggered was an explosive synthesis laid down by Gromov, represented by two closely related theories which have been burgeoning for the last 40 years: the theory of hyperbolic groups (and associated theories of nonpositively curved groups of various flavors); and the theory of quasi-isometric classification of finitely generated groups.

A third explosion, arising from Milnor's work mentioned above and therefore also part of the line of descent from Dehn's Algorithm, is Grigorchuk's example of a finitely generated group of intermediate growth, meaning growth that lies strictly between polynomial and exponential. This example led to many followup questions and associated theories, such as the theory of automata groups (not to be confused with automatic groups). This theory (and other ground breaking counterexamples) has also motivated the art of "constructions", which I think of as the use of ideas surrounding small cancellation theory and techniques developed by Grigorchuk, Rips, and others to construct new groups that confound expectations and break conjectures. There is also a certain overlap between this theory and the Novikov--Boone work mentioned in the answer of @user1729.