TITLE: Theorems with the greatest impact on group theory as a whole
QUESTION [28 upvotes]: 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?

REPLY [6 votes]: 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.